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© 2000-2023   Gérard P. Michon, Ph.D.

Divergent Series Redux
( A Progress Report )

All [ Hardy's ] books gave him some degree of
pleasure, but this one, his last, was his favourite.
[ ... ]  He believed in its value (as well he might).
Preface, by  John E. Littlewood, to the last book of
G.H. Hardy (1877-1947):  "Divergent Series" (1949)
 
Being eternal,  Logic can be patient.   Pierre Duhem (1913)

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Types of Series and Types of Summability  by  C.N. Moore  (April 1931).
 
On Riesz Summability and Summability by Dirichlet's Series  C. T. Rajagopal
American Journal of Mathematics, 69, 2 (April 1947) 371-378  (1946-08-02).
 
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Encyclopedia of Mathematics :   Summation methods   |   Semi-continuous summation methods
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Former IAS member Carl Bender received the  2017 Dannie Heineman Prize for Mathematical Physics.
 Carl Bender at the blackboard
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Math History: Infinite series (1:11:00)  by  Norman J. Wildberger  (UNSW, 2011-06-06).
Adding past infinity (0:47)  &  Taming infinity (1:34)  by  Henry Reich  (Minute Physics, Summer 2011).
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1+2+3+4+5+... = -1/12  (again)  by  Ed Copeland  &  Tony Padilla  (Numberphile, 2014).
A take on  1+2+3+4+5+... = -1/12  by  Edward Frenkel  (Numberphile, 2015-01-11).
Ramanujan: Making sense of 1+2+3+4+5+... = -1/12  by  Burkard Polster  (Mathologer, 2016-04-22).
The Remarriage of Pure Mathematics & Theoretical Physics (1:00:47)  by  Dave Morrison  (Aspen, 2015-08-27).
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Divergent Series Redux
Summation Theory

In 1713, Leibniz said that all divergent series can be evaluated.  Can they?

A series is just an ordered sequence of infinitely many terms which we seek to add together  (order matters).  A partial sum of such a series is obtained by adding its first terms.  If partial sums have a limit, the series is said to be convergent and that limit is called the sum of the series.  Non-convergent series are called divergent.  The notion of sum can be extented to some divergent series  (either stable or unstable in the sense discussed below).


(2012-07-24)   Sum of a  Divergent  Geometric Series
How can a definite  sum  be assigned to a divergent series?

Pour moi, j'avoue que tous les raisonnements fondés sur les séries
qui ne sont pas convergentes
  [...]  me paraîtront très suspects, même
quand les résultats s'accorderaient avec des vérités connues d'ailleurs.

D'Alembert, 1768   [Opusc. math., 5, p. 183 ]

Analytic continuations  can make sense of some  divergent series  in a consistent way.  Consider the following  classic summation formula  (attributed to  Eudoxus)  for the  geometric series,  which converges when the  common ratio  z  is small enough in  magnitude  (it diverges when  |z| > 1 ) :

1  +  z  +  z +  z +  z +  ...  +  z +  ...   =   1 / (1-z)

The right-hand-side always makes sense, unless  z = 1.  It's tempting to equate it  formally  to the left-hand-side, even when that series diverges!

This viewpoint belongs to a  consistent  body of knowledge which is still not mature, in spite of its exploration by generations of great mathematicians.  The following monstrosities do make sense as "sums" of  divergent series :

1  +  2  +  4  +  8  +  16  +  ...  +  2 +  ...   =   -1
1  +  3  +  9  +  27  +  81  +  ...  +  3 +  ...   =   - ½

In rings, whenever both sides of such equations are defined, they are necessarily equal.  In  p-adic arithmetic,  for example,  the above geometric series  converges  (only)  when  z  is an integer which is divisible by the modulus  p  (use p=2 and p=3, respectively, for the above two examples).

The following series thus converges in dyadic, triadic or hexadic  integers:

1  +  6  +  36  +  216  +  1296  +  ...  +  6 +  ...   =   - 1/5

The respective sums can be given "explicitly" in each of those three cases:

Dyadic:   ...1001100110011001100110011001100110011001100110011
Triadic:   ...012101210121012101210121012101210121012101210121
Hexadic:   ...1111111111111111111111111111111111111111111111111111

Convergence in some  ad hoc  realm is a comforting luxury which is rarely available.  There's no such thing in the case of what's arguably the  simplest  divergent series  (Grandi's series, 1703)  whose  long history  started with an incidental remark of  Leibniz  in 1674  (De Triangulo Harmonico) :

-  1  +  1  -  1  +  ...  +  (-1) +  ...   =   ½

The best way to index the successive terms in Grandi's series is to start with  n = 1.  That makes it a  multiplicative  sequence.  The correct way to extend any multiplicative sequence down to the index  n = 0  (when needed)  is to assign it a value of zero there.  Thus,  I consider dubious the popular use of  A033999  as a representation of Grandi's series.  I'm adamant about using instead the following sequence  (whose closed form on the left-hand-side depends on the  fact  that the zeroth power of zero is unity).

0n - (-1)n   =   0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1,  ...   (A062157)

Because  Grandi's series is  stable  (as a  geometric series  whose ratio isn't unity)  the leading zero is irrelevant to the summation.  This indexation which makes Grandi's sequence multiplicative also makes its  Dirichlet inverse  multiplicative  (and characterized by values at powers of primes):

0, 1, 1, -1, 2, -1, -1, -1, 4, 0, -1, -1, -2, -1, -1, 1, 8, -1, 0,  ...   (A067856)

  •   a (2n )   =   2n-1
  •   If  p  is an odd prime,   a (p)  =  -1   and  a (pn )  =  0   for  n > 1.

The  convolutional inverse  of Grandi's series  (with its leading zero)  is a finite series with just two nonzero terms:

1 + 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + ...   =   2     (A019590)

That's yet another way to show that  Grandi's series  has a sum of  ½.

History of Grandi's series   |   Divergent geometric series   |   1 + 2 + 4 + 8 +...   |   1 - 2 + 4 - 8 +...
Videos :   Convergence with respect to the dyadic metric:  Invent math  by  Grant Sanderson (Aug. 2015).
Visualizing analytic continuation  by  Grant Sanderson (2016-12-09).


(2018-05-23)   Definitions and notations  (inherited from tradition).
The equal sign may be used to say that a scalar is the  sum  of a series.

A  (formal)  series  is just a  vectorial object  consisting of an infinite sequence of scalar coefficients  (called  terms).

The formal series of general term  an  can be denoted:

Sn  an

summation method  (also  summation mapping  or  summation,  for short)  is a recipe which assigns a scalar value  (called  sum)  to some  series  in such a way that the newly defined  sum  of a series with only finitely many nonzero terms is the ordinary (finite) sum of those nonzero terms.

The sum of a series using the summation method  M  is denoted with the name of  M  above the sigma sign  (or as a superscript).  If the summation method is clear from the context,  square brackets may also be used  (such bracket thus turn a series into a scalar):

M  
[ Sn  an ]   =   Sn  an

For typographical simplicity and for compatibility with traditional notations,  brackets may be dropped in equations when it's clear that summation is intended  (because the vectorial object itself wouldn't make sense in the context).  That's so when a series is  equated  to an explicit scalar.  Example:

[ Sn  an ]   =   Sn  an

In particular,  the above requirements for  any  summation method imply:

0   =   Sn  0

Normally,  the index  n  runs from  0  to infinity,  through the  set of natural integers   N = {0,1,2,3,...}.  It's sometimes convenient to the use the  set of counting numbers  instead N* = {1,2,3,4,...).  When we do so,  we signal it with a star after the index  (this is not a standard notation).  In other words:

Sn*  an   =   Sn  an+1

That's just an equivalence of notation,  without a deep mathematical content:  It just says that a sequence of scalars remains the same no matter how we index it.  The above just expresses the identity between two ways to denote the same formal vectorial object irrespective of any property attached to it  (including summability).  Formal series are just vectors with a  (countable)  infinity of coordinates.  Such vectors are equal if and only if all of their coordinates are pairwise identical,  as is the case with only finitely many  dimensions.  Another notation for that same  vectorial  identity is:

S   an     =     S   an+1
n=1 n=0

On the other hand,  if we put  anything  above the sigma sign,  then some kind of summation is implied.  This could be via one of the highbrow methods discussed on the present page  (or elsewhere)  or by straight  convergence  (when applicable)  in which case the  infinity symbol  (¥)  can be used.  That way,  we retrieve a familiar notation exemplified by:

¥   1 / 2n     =    2
S
n=0

Thus,  the infinity symbol takes the place we may use to identify other summation methods which gives new meaning to the nice title of a relevant short video by  Henry ReichAdding Past Infinity.

Reich was just lucky that the  divergent  example he picked happened to be a  stable  series.  If we do away with the above sigma notation  (as we often will)  the  stability  property Reich took for granted says that:

a0 + ( a1 + a2 + a3 + ... )   =   a0 + a1 + a2 + a3 + ...

For  some  series this isn't so.  Such  unstable  series include some  physically relevant ones  which motivated Reich's  next episode,  like:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + ...

That series isn't stable  (as we shall see)  but we can still uncover a sum for it  (namely, -1/12).  That sum is  modified  by adding finitely many zeroes  "in front of it",  but in a  consistent way  which relies only on  linearity.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + ...   =     - 1/12
0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ...   =     + 5/12
0 + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + ...   =   + 23/12
0 + 0 + 0 + 1 + 2 + 3 + 4 + 5 + 6 + ...   =   + 53/12

To navigate this foggy land,  we can only rely on precise instruments...


(2012-07-29)   Desirable Properties of General Summation Methods
Two  consistent  methods give the same result when both are defined.

The mapping which assigns a sum to a series ought to possess a few desirable properties,  but we must be prepared to abandon some of these  (except linearity)  to allow the most powerful summation methods  (e.g.,  those which can handle interesting  unstable  divergent series).

1 - Linearity :

Let's split  linearity  into two separate requirements :

  • Scalability :   Sn  l an   =   l  Sn  an
  • Additivity :   Sn  ( an + b)   =   Sn  an   +   Sn  bn

Linear summation methods are thus linear forms defined over the  vector space  of  formal series  (in any vector space, a  form  is generally defined as a function which takes a vector to a scalar).

2 - Stability  (a.k.a.  Translativity  or  Shift-Invariance ) :

Stability  is the following  nontrivial  rule,  inspired by finite sums:

Sn  an   =   a0  +  Sn an+1

That's also known as  shift-invariance  (explicitely required for summation by  Banach limit,  when the sequence of partial sums is  almost convergent ).

An uresolved question is whether a shifted stable series is necessarily stable again.  If that's so,  the above rule says that you can freely shift such a series to the right or to the left without compromising summability and that the sum is only affected as would be naively expected  (there's no reason to distinguish between right or left stability for linear  summation methods).

The stability property is  treacherous.  It turns out that it's really a property of the  series  itself and not of the  summation method  being used.

A great deal of confusion comes from the fact that many summation methods are valid  only  for stable series  (summation by convergence being the most obvious example).  In such a case  (and  only  in such a case)  we say that the summation method itself is  translative.

As we shall see later on,  1 + 1 + 1 + 1 + 1 + ...  =   -½   If that series was stable, of sum x, the equation  x = 1+x  would hold.  It ain't so...  There's an elementary proof that the series can't possibly be stable under any linear summation method.  (HINT:  If x is the sum of the original series and y is the sum of the shifted series,  then  x-y = 1.)

3 - Regularity :

A summation method is said to be  regular  if it assigns to any  convergent  series its  ordinary  sum  (i.e., the limit of its partial sums).

By  definition,  all summation methods are  finitely regular  (i.e.,  they all give the same result for any series with finitely many nonzero terms)  but some of those are not regular  (HINT:  consider the  Riemann series theorem).

4 - Multiplicativity :

The  formal series  form not only a vector space but also a  ring.  We'd like the summation of series to respect that structure and may want the sum of a  Cauchy product to equal the product of the sums of its two factors, namely:

( Sn  a)  ( Sn  b)   =   Sn  ( S k ≤ n  an-k bk )

The ordinary summation of convergent series is certainly multiplicative in that sense.  So is the Cesàro-summation of Cesàro-summable  series:  One example involving  divergent  (but  Cesàro-summable)  series is the following pair of series,  (the second one is the Cauchy-square of the first):

-  1  +  1  -  1  +  1  -  1  +  1  -  ...  +  (-1) +  ...     =   ½
-  2  +  3  -  4  +  5  -  6  +  ...  +  (-1)n (n+1)  +  ...   =   ¼

Unstable series need not be so well-behaved.  Here's an example of a related series whose squared sum  isn't  the sum of its Cauchy square:

1  +  1  +  1  +  1  +  1  +  1  +  1  +  ...  +  1  +  ...   =   -1/2 
1  +  2  +  3  +  4  +  5  +  6  +  ...  +  (n+1)  +   ...   =   -1/12

5 - Semi-Continuity :

If the limit of  fn (x)  is  1   (unity)  for every index  n,  as  x tends to  x in some limiting process, then we have, for the same limiting process:

Sn  a   =   lim xSn  an fn (x)  ]

This may be summarized:  The sum of the limits is the limit of the sums  (the space of  formal series  is endowed with the Tychonoff topology).

As every student of  Analysis  ought to know,  that's not so in the restricted realm of  convergent series,  as the series formed by the limits  (of the corresponding terms of in a sequence of convergent series)  is not necessarily a convergent series.

Historically, the desired continuity of the summation in the realm of divergent series was rarely evoked explicitly but it has almost always been  assumed  by everyone...
 
The traditional parlance, since 1907, is to call the coefficients  fn (x)  convergence factors  whenever they allow the above right-hand side to make sense in a relevant neighborhood of the limit point for x.

It was Charles N. Moore  (1882-1967)  who coined that locution in 1907.  It appears in the title of his Harvard doctoral dissertation  (1908).  The French equivalent  (facteur de convergence)  was used systematically in a new chapter (VI) of the second edition (1928) of Borel's  " Leçons sur les séries divergentes " (1901)  without any indication of its origin  (later claimed, in print, by Moore himself).

  Charles N. Moore
Charles N. Moore

For regular summation methods, whenever the series involved on the right-hand-side are convergent and their sums have a finite limit, that limit is the same for all factors of convergence  (this is left as an exercise for the reader)  and can thus be used as a definition of the sum of the series on the left-hand-side if it happens to be divergent.  What the semi-continuity of a summation method ultimately states is that the above equation also holds when sums of divergent series are involved on the right-hand-side...

Moore's Theorem   (2016-05-05)

The key remark, which I like to call  Moore's theorem,  is that two different sequences of  convergence factors  which "work" will give the same result.  That's to say that the following equation holds whenever both sides make sense with two sequences  f  and  g  whose terms all have a limit of  1.

lim xSn  an fn (x)  ]   =   lim xSn  an gn (x)  ]

In 1900,  Alfred Pringsheim  defined a doubly-indexed sequence as having a limit  L  if its terms approach  L  when both indices tend to infinity.

Pringsheim double limit (1900)   |   Alfred Pringsheim (1850-1941)


(2018-05-26)   Stability of the geometric series.
The geometric series whose  common ratio is unity cannot be stable.

The geometric series of  common ratio  1  is  not  stable,  but  all  other geometric series can be assigned a precise stable sum.  Let's prove that:

We assume a linear summation method powerful enough to assign a finite sum  S  to the geometric series at point  z.  Which is to say:

1  +  z  +  z +  z +  z +  ...  +  z +  ...   =   S

Stability would then be expressed by the following relation,  which would seem obvious to the uninitiated  (it's what's often implicitly used as a preliminary step in the derivation of the sum of  convergent  geometric series).  Actually,  that's what we want to prove  or disprove :

0  +  1  +  z  +  z +  z +  ...  +  z +  ...   =   S  (?)

By definition, the sum of a series with finitely many nonzero terms is easy:

1  +  0  +  0  +  0  +  0  +  0  +  0  +  0  +  ...   =   1

As  linearity  is assumed,  we may subtract that from our original equation:

0  +  z  +  z +  z +  z +  ...  +  z +  ...   =   S-1

Also by linearity,  we're allowed to scale that by a factor of 1/z  (if z is nonzero)  and we obtain the sum of the series we were after:

0  +  1  +  z  +  z +  z +  ...  +  z +  ...   =   (S-1)/z

So,  excluding the trivial case  z=0,  the stability of the geometric series of common ratio  z  is  equivalent  to the equation:

S   =   (S-1)/ z

This equation has no solution when  z=1,  which goes to show that the geometric series of common ratio unity is not stable.  Otherwise, it implies:

S   =   1 / (1-z)

That's the only  stable  sum a  linear summation method  can assign to the geometric series of common ratio  z.

An  ad hoc  requirement for  linear summation methods  could be to impose the stability of geometric series of nonunity ratio.  We've refrained from adding that to our  basic list.  Continuity seems far more fundamental,  albeit less elementary,  as presented  below.


(2020-05-13)   Stability of any series with vanishing terms.
Any series whose terms tend to zero is stable.

This applies even when such a series doesn't converge  (one example being the  harmonic series).  This is so because the  difference  between the series and the series obtained by shifting it to the right  (by one position or several, with zero padding at the beginning)  is a convergent series of zero sum...

Indeed, that difference is just a  telescoping  series whose partial is equal to a single term of the original series  (or the sum of several such terms if we shift by several positions).  Since we assume that those terms tend to zero,  the limit of the partial sums is zero;  the difference is thus a convergent series of zero sum.    QED


 Gerard Michon  Augustin Cauchy (2018-06-23)   The Cauchy product of two series.
It's a stable series when at least one of the factors is stable.

A couple of classical theorems pertain to the convergence and/or summability of the Cauchy-product of two series which have prescribed properties.  (In all of these cases,  the sum of the Cauchy-product is the product of the sums.)

The Cauchy product of two  unstable  summable series can be a summable series whose sum  isn't  the product of the two sums  (cf.  example  above).

This doesn't seem to happen when at least one of the two series is stable.

It can be useful to think of the Cauchy product as an  additive convolution  to be contrasted with the  mulplicative convolution  corresponding to the  Dirichlet product  described in the next section.  They can be defined using very similar notations.  Compare the following definition of the Cauchy product to the parallel definition given  below  for the Dirichlet product.

å   f × g   =   å    å   f (d)  g (n-d)    =    å    å   f (i)  g (j)    =    å   f   å  g  
    n ≥ 0 d ≤ n   i ≥ 0 j ≥ 0  
 

In both cases,  we may consider what happens when all series involved are  absolutely convergent.  In that case,  all terms can be freely permuted and the above equality is established by remarking that every term  f (i) g (j)  appears once and only once on either side.  In other words,  the  sum of the product is the product of the sums.  That nice statement is not necessarily true in general  (we've already seen a  counter-example).

Summing by Convolutional Inversion  (2019-07-24 15:15 PDT)

Over any  unital ring,  the formal series form a unital ring  (whose unity is the series whose only nonzero term is a leading term equal to unity).  Any series whose leading term is unity has an inverse with respect to the Cauchy product,  which is often called its  convolutional inverse.

When that inverse is summable and has an  invertible sum,  it's tempting to believe that the inverse of that is the sum of the original series.  In some cases,  we can prove that to be true...

The simplest example is  Grandi's series with a leading zero  whose convolutional inverse is a series consisting of just two nonzero terms:

0  +  1  -  1  +  1  -  1  +  1  -  1  +  1  -  1  +  ...   =    1/2
1  +  1   =   2

One related example which  doesn't  work is the following pair of series which are convolutional inverses of each other  (watch  leading zero).

0  +  1  +  1  +  1  +  1  +  1  +  1  +  1  +  1  +  ...   =   -3/2
1  -  1   =   0

Cauchy product   |   Mertens theorem   |   Cesàro's theorem
 
"Sur la multiplication des séries" by Ernesto Cesàro.  Darboux's Journal, 14, pp. 114-120  (1890).


 Gerard Michon (2019-05-15 02:30 PDT)   Dirichlet Product  of Two Series
The sum of the product is the product of the sums.  Or is it?

When all series involved are  absolutely convergent,  their terms can be freely permuted and the above statement is established by remarking that every term  f (i) g (j)  appears once and only once in either of the following summations:

å   f *g   =   å    å   f (d)  g (n/d)    =    å    å   f (i)  g (j)    =    å   f   å  g  
    n ≥ 1 d | n   i ≥ 1 j ≥ 1  
 

Dirichlet Summation Method :

Any series whose first term is nonzero has a  Dirichlet-inverse.  If that Dirichlet-inverse is summable,  then the series is said to be Dirichlet-summable and its sum is the reciprocal of the sum of its Dirichlet-inverse.

That implies a sum of  -2 for the  Moebius series   å m(n)  whose inverse is:

1 + 1 + 1 + 1 + 1 + ...   =   -1/2

A less chancy example would be the Dirichlet-inverse of  Grandi's series:

1 + 1 - 1 + 2 - 1 - 1 - 1 + 4 + 0 - 1 - 1 - 2 - 1 - 1 + 1 + 8 - ...   =   2

Lambert series   |   Mock modular form  (Appell-Lerch sum)


 Gerard Michon (2019-07-05 17:45 PDT)   Infinite Products
Exponentials of infinite series  seem  less ambiguously defined.

The situation is apparently similar to the way ambiguities are lifted when going from the formula of  Roger Cotes  to  Euler's formula.  However,  the full formalism of  Riemann sheets  cannot be bypassed in general.


 Gerard Michon (2018-07-03)   Series whose terms are themselves sums of series.
Countable additivity  yields two distinct ways to obtain such sums.

One special case can be derived from the following  termwise  equality.

Si Sj ai bj   =   Si [ Sj ai bj ]   =   Si ai [ Sj bj ]   =   [ Sj bj ] Si ai

The general relation is guaranteed if we have  absolute  convergence:

[ Si Sj aij ]   =   [ Sj Si aij ]

Otherwise,  that only gives a questionable  hint.  One such hint would be that the series whose  n-th  term is  s0(n),  the number of divisors of n,  has a sum of  ¼  (HINT:  All  decimated series  S0(k.k-1)  have the same sum).

Likewise for any other  divisor function  s,  where  s(n)  is defined as the sum of the s-th powers of all the positive divisors of n:

1+1+ 1+1+ 1+1+ ...   =  S0(1,0)  =  -1/2
0+2s+ 0+2s+ 0+2s+ ...   =  2s S0(2,1)  =  -2s/2
0+0+3s+ 0+0+3s+ ...   =  3s S0(3,2)  =  -3s/2
0+0+0+4s+ 0+0+ ...   =  4s S0(4,3)  =  -4s/2
0+0+0+0+ 5s+0+ ...   =  5s S0(5,4)  =  -5s/2
0+0+0+0+ 0+6s+ ...   =  6s S0(6,5)  =  -6s/2
...
 bar
ss(1)+ ss(2)+ ss(3)+ ss(4)+ ss(5)+ ss(6)+ ...  =   Bs+1 /(2s+2)   =   - z(-s)/2

The following example shows that this cannot be applied blindly.  It's useful to investigate conditions  (besides absolute convergence)  which allow this.

-1+1/2+ 1/22+1/23+ 1/24+1/25+ ...   =  0
0+-1+1/2+ 1/22+1/23+ 1/24+ ...   =  0
0+0+-1+1/2+ 1/22+1/23+ ...   =  0
0+0+0+-1+1/2+ 1/22+ ...   =  0
0+0+0+0+-1+1/2+ ...   =  0
0+0+0+0+0+-1+ ...   =  0
...
 bar
-1-1/2- 1/22-1/23- 1/24-1/25- ...   =  -2

 Come back later, we're
 still working on this one...

 Jakob Bernoulli  Ka-mon

(2018-07-16)   Original  Bernoulli Numbers   (c. 1689)
Also in the posthumous legacy of  Seki (1712).

The symbol  Bn  denotes the  n-th  Bernoulli number,  as introduced  above  in terms of the  Zeta function  (yielding unambiguously  B= ½).

Bernoulli Numbers :   A164555 / A027642  =  A164555 / A141056
n 0123456 7891011121314...
 B 11/21/60-1/3001/42 0-1/3005/660 -691/273007/8...

Many aspiring algebraists,  like my younger self,  have rediscovered those in the elementary context of  Faulhaber's formulaBernouilli numbers  are often found in  Number Theory  and play a key rôle for  infinite series  via the  Euler-Maclaurin formula  and  Ramanujan summation.  They're best defined through their  (exponential)  generating function :

z / (1 - e-z )   =   Sn  Bn zn / n!       (convergent for   | z |  <  2p )

Everybody agrees on the values of Bernoulli numbers for even values of the index  n  (some authors used to report only even values by halving the index: A000367/A002445).  However,  the sign of the only nonzero odd value  ( B) remains controversial.  The alternate value for  B1  appearing in either  A027641/A027642  or  A027641/A141056)  was endorsed  de facto  by the NIST (1964) and Wolfram Research  (but not by Bernoulli,  Seki,  Terry Tao...  or my adamant self).
 
The confusion arises from a competing equally-simple generating function,  obtained by changing the sign of z.  The difference between the two generating functions is given by a remarkable easy-to-prove identity,  which shows that both only have one nonzero odd coefficient and that their even coefficients coincide:

z / (1 - e-z )   -   z / (ez - 1)     =     z

Whenever possible,  it's probably best not to take either convention for granted and give explicitly the first terms of an expansion followed by a general expression involving only Bernoulli numbers of  even rank  (see example below).

Bernoulli polynomials (Numericana)   |   Bernoulli numbers (Weisstein)


(2012-08-03)   Summation as a covector  (i.e., a continuous linear form).
(In infinitely many dimensions, linear forms could be discontinuous.)

The  formal series  (irrespective of their convergences)  form a  sequence space.  Using Dirac's notation, a formal series can be described as a  ket :

Sn  an   =   Sn  an | n >   =   | y >

continuous  and  linear  summation method is a  bra  < s |   Such a  bra  is a member of the  continuous dual  of the aforementioned sequence space  (the  algebraic dual  consisting of  all  linear forms is much larger, by the Axiom of Choice).

Formally,  a  regular summation can only be equal to the following  bra :

s |   =   Sn  < n |

However, that expression is of no practical value, unlike some of the following methods which describe  < s | better.

Let's define the  (non-invertible)  shift  operator  Û  via:

Û | n >   =   | n+1 >

If  | y >  is  stable,  we have:   < s | y >   =   < s | Û | y >

Duality:  The smaller the subspace,  the bigger its dual.

In general,  the  [continuous]  dual  of a  [topological]  vector space  consists of all  [continuous]  linear  functions defined on it  (two such functions being equal if and only if they have the same value everywhere).

The dual of any linear space  E  so defined is a well-defined set  (a subset of the  Cartesian square  E)  which is itself a linear space.

In this work we consider only  continuous  duals and denote  E*  the  [continuous]  dual of  E.

Furthermore,  our attention is restricted to the case when  E  is some subspace of the sequences of scalars.

 Come back later, we're
 still working on this one...

Duality


 Louis Augustin Cauchy 
 1789-1857 (2012-08-10)   Summation by Convergence
The only method  Cauchy (1789-1857)  would ever recognize.

The sequence of the  partial sums  of a series is the sequence whose term of index  n  (usually starting at  n = 0)  is obtained by adding the finitely many terms whose index in the series does not exceed  n.

If that sequence of partial sums  converges  to a limit  S,  the series  itself  is said to be convergent  (of  sum  S).

For convergent series  (at least)  we make no typographical distinction between a series and its sum.  Thus, we express the above as:

Sn  an     =     [  Sn  an  ]     =       (  m   an  )     =     ¥   an
lim S S
m ® ¥ n = 0 n = 0

The last equation merely expresses the conventional notation for the quoted limit of partial sums.  Nothing else.

When that limit doesn't exist,  Cauchy  argued that the leftmost expressions don't make sense.  Two generations before him, the great  Euler  had taken the opposite view,  rather freely,  with great success  (Ramanujan  would do the same much later).  Cauchy simply had deep concerns that the lack of explicit rules for manipulating divergent series made any argument based on them doubtful at best.  So he decided to rule them out!

The pendulum has swung back.  Nowadays, divergent series can be handled with complete analytical rigor.  Both  Cauchy and Euler would be pleased...

The following sections will trace the historical path away from Cauchy's strict point of view, then break free completely...

Summation by convergence is just the simplest  regular  summation method, among mutually  consistent  ones which apply to divergent series as well.

The other such methods, including those described below, can be classified into two broad groups:

  • Summations by means.
  • Summations by convergence factors.

Sometimes, those two types of approaches are known to be equivalent.


(2012-09-09)   The Functional Analysis Approach
How to extend a continuous summation method to a larger domain.

By definition, an  absolutely regular  summation method is a continuous functional defined for every absolutely convergent series which coincides with the ordinary summation by convergence.  An  absolutely regular  summation method need not be regular.

For example, a permutation of the indices always leaves unchanged the sum of any absolutely convergent series.  However,  the  Riemann series theorem  says that it  may change  arbitrarily the sum of other convergent series and, thus, induce a non-regular summation method.

 Come back later, we're
 still working on this one...

Hahn-Banach extension theorem


 Leonhard Euler
 1707-1783 (2012-08-09)   Euler Summation   (1746)
The earliest method of  summation by convergence factors.

Euler introduced the following  definition  (expressed in our  notations):

[  Sn  an  ]     =       (  ¥  )
lim S  an xn
x ® 1- n=0

Apparently,  this is the first explicit use of the  postulated  continuity of summation.  For example, Euler famously derived the following sum:

-  2  +  3  -  4  +  5  -  6  +  ...  +  (-1)n (n+1)  +  ...   =   ¼

As the limiting case of this convergent summation, as z tends to 1 on [0,1[

-  2 z  +  3 z2  -  4 z3  +  ...  +  (-1)n+1 (n+1) z n  +  ...   =   1 / (1+z)2

Note,  however,  that Euler's method isn't powerful enough to handle the nonalternating case  (corresponding to  z = -1)  which is one of our favorite examples of an  unstable  series.  This much can be construed as a consequence of the following theorem:

 Gerard Michon

Any Euler-Summable Series is Stable.  (2018-05-31)

Proof:   This is a straight consequence of the stability of convergent series  (as implied by the notation we used for Euler's above definition,  only convergent series appear in the right-hand-side).  Indeed,  the limit of the sum of  two  sequences is the sum of their respective limits, when both exist.

Abel's limit theorem (1826)


(2019-06-03)   Cesàro Summation   (1890)
Summation by averaging.

 Gerard Michon

Any Cesàro-summable series is stable.  (2019-06-03)

Proof:  

Cesàro summation (1890)   |   Ernesto Cesàro (1859-1906)   |   Silverman-Toeplitz theorem
 
"Sur la multiplication des séries" by E. Cesàro.  Darboux's Journal, 14, pp. 114-120  (1890).

 Emile Borel
Emile Borel

(2012-07-29)   Borel Summation   (1899)

By Euler's integral of the second kind, the following identity holds for any nonnegative integer  n :

ò0¥  tn/n!  e-t dt    =   1

Therefore, the following is a trivial equality between  identical  series :

Sn  a    =    Sn  ò0¥  a tn/n!  e-t dt

In 1899,  Emile Borel (1871-1956)  thus proposed to  define  the left-hand-side  (which could be a divergent series)  by equating it to the right-hand-side of the following formula, at least when the new series that is so formed is a  power series  of  t  with an infinite radius of convergence,  which makes the resulting (improper) integral converge:

Sn  a    =    ò0¥  ( Sn  a tn/n! )  e-t dt

The bracketed series on the right-hand-side clearly stands a better chance of converging than the series on the left-hand-side.

For example, in the case of the geometric series  (an = z)  the above integrand becomes  exp [(z-1)t]  which makes the integral on the right-hand-side converge when the  real part  of  z  is less than one.  We thus have convergence of the Borel formula for half the plane, whereas the left-hand-side merely converges on a disk of unit radius.


Borel summation, however, is best understood as a way to obtain the sum of a series from the sum of another, even if the latter is not convergent...

For example, armed with the formula for the sum of a geometric series  (convergent of not)  we can use the Borel summation to make mincemeat of the following series which Euler  (E247, 1746)  called  hypergeometric series of Wallis  (the name is obsolete;  I recommend the unambiguous name  Euler-Gompertz series).  He evaluated it in half a dozen distinct  consistent  ways:

1  -  1  +  2  -  6  +  ...  +  (-1)n n!  +  ...   =   ò0¥  ( 1 + t ) - 1  e-t dt

That's equal to   d   =   -e Ei(-1)   =   0.596347362323194074341078499... That value is the  Euler-Gompertz constant (A073003)  which is named after  Benjamin Gompertz (1779-1865).  The symbol  d  was introduced in 2009,  by  Alexander Ivanovich Aptekarev  (b. 1955,  Ph.D. 1981).

Some authors call this  "Euler's series".  To Euler,  hypergeometric numbers  were just what we now call factorials  (ironically, the latter term had been coined by Wallis himself,  in 1655).  The locution "hypergeometric series" is now reserved for a different creation of Gauss (1812).

The nonalternating series involves a  Cauchy principal value :

1  +  1  +  2  +  6  +  24  +  ...  +  n!  +  ...   =   ò0¥  ( 1 - t ) - 1  e-t dt

The numerical value of that is  0.76521907949...

Borel summation   |   1-1+2-6+24-120+...   |   Euler-Gompertz series  by  Adriána Szilágyiová  (2016-11-21).
Euler's Constant:  Euler's Work and Modern Developments  by  Jeffrey C. Lagarias  (2013-10-25)
How Euler Did It:  Divergent Series  by  Ed Sandifer   (MAA Online, June 2006)


(2012-08-21)   Nørlund summation:  Linear, stable & regular   (1919)
All  Nørlund means  are  consistent  with one another.

Following  Niels E. Nørlund  (1919)  let's consider an infinite sequence of complex ponderation coefficients, the first of which being nonzero:

c0 ¹ 0 , c1 , c2 , c3 , c4 , ... , cn , ...

Let's call  C  the sequence of the partial sums of the corresponding series:

Cn   =   c0 + c1 + c2 + c3 + c4 + ... + cn

We impose the so-called  regularity condition :

  • The positive series of term  | cn / Cn |  is  convergent.

In particular, coefficients in  geometric progression  are thus ruled out,  if the common ratio is greater than 1.  So are coefficient sequences that grow faster than such a geometric sequence.

For any series  an  with partial sums   A n  =  a0 + ... + an   we define:

A'n   =   ( c0 A n  +  c1 A n-1  +  c2 A n-2  +  ...  +  cn A 0 ) / C n

This expression is called a  Nørlund mean.  If  A'n  tends to a limit  S  as  n  tends to infinity, then  S  is called the  Nørlund-sum  of the series  an  or, equivalently, the  Nørlund-limit  of the sequence  A n .

Remarkably, the value of  S  doesn't depend on the choice of the sequence of coefficients  (with the above  regularity  restrictions).

 Come back later, we're
 still working on this one...

Nørlund means (1919)   |   Niels E. Nørlund (1885-1981)
Sur une application des fonctions permutables  (N.E. Nørlund, 1919)
On Convergence Factors for Series Summable by Nørlund Means   by  Charles N. Moore   (1935)


(2012-08-26)   Hausdorff summation:  Linear, stable & regular   (1921)
Consistent with  Nørlund summation.

The question of the consistency of Nørlund and Hausdorff methods was raised by E. Ullirich and it was answered  (in the affirmative)  by  W. Jurkat and A. Peyerimhoff,  in 1954.

 Come back later, we're
 still working on this one...

Hausdorff summation   |   Felix Hausdorff (1868-1942)
The consistency of Nørlund and Hausdorff methods   by  W. Jurkat and A. Peyerimhoff  (1954)

 Niels Abel
Niels Abel
 
 Niels Henrik Abel  
 1802-1829
(2012-08-09)   Abel Summation   ()

The divergent series are the invention of the devil,     
and it is a shame to base on them any demonstration
  ... 
Niels H. Abel  (1802-1829)     

The  summation method  proposed by Abel is:

 Come back later, we're
 still working on this one...

Abel summation   |   Abelian and tauberian theorems
Tauberian conditions (1897)   |   Hardy-Littlewood tauberian theorem (1914)   |   Alfred Tauber (1866-1942)
 
Littlewood's Tauberian theorem (3:07)  by  Freeman Dyson  (Web of stories, 1997).


(2012-08-09)   Lindelöf Summation   (1903)

 Come back later, we're
 still working on this one...

Lindelöf summation (Wikipedia)   |   Lindelöf summation method   |   Ernst Lindelöf (1870-1946)


(2012-08-28)   Mittag-Leffler Summation Method   (1908)

As he was reflecting on the summation of the geometric series in 1908,  Mittag-Leffler  proposed the most widely applicable definition he could think of, at the time  (using the  Gamma function) :

Sn  an    =       (  Sn  
an
vinculum
G(1+en)
 )
lim
e ® 0+

For the geometric case  (an = zn )  the right-hand-side converges except when  z  is a real greater than or equal to 1.

More generally, convergence normally occurs on a  Mittag-Leffler star  consisting of all points of the complex plane, except the  shadows  of singular points  (i.e.,  wherever a singular point exists on the straight ray from zero to that point).

Mittag-Leffler summation method   |   Mittag-Leffler star   |   Gösta Mittag-Leffler (1846-1927)

 Georges Valiron
Georges Valiron, 1932
 

(2021-07-27)   Valiron Summability   (1917)

Georges Valiron (1884-1955)  was notably the thesis advisor of  Laurent Schwartz (1915-2002).

Georges Valiron (1884-1955)
 
Georges Valiron, Rendiconti di Palermo, 42, 267-284 (1917).
On the Summability of Series by a Method of Valiron  by  J. M. Hyslop  (1935-10-14).


(2016-05-08)   Generalized Summation Methods   (c. 1919)
Generalizations and early attempts at defining the sum of divergent series.

In March 1918 at the University of Washington,  Lloyd Leroy Smail (1888-1955)  defined the sum of a series by the following equation, whenever the right-hand-side makes sense for a sequence of functions  f(m,x)  which all tend to  1  as  m  tends to infinity and  x  tends to a given limit point  x:

¥   [     (   m   )  ]  
å   an   =    lim lim å   a fn (m, x)
n = 0 x m ® ¥ n = 0

He pointed out that many previously devised summation methods merely reduce to different choices for  f  in the work of different authors, including:

WhoWhen  f (m,x)  x0Notes
Cauchy1 
Eulerxn1-x < 1
Hölder1882 (1 - n/(m+1))kk real constant

  • Euler.
  • Cesàro.
  • Hölder.
  • Riesz.
  • de la Vallée-Poussin.
  • Plancherel.
  • Leroy.
  • Borel...

 Come back later, we're
 still working on this one...

Louis Silverman,  On Various Definitions of the Sum of Divergent Series.  (Dissertation, 1910).
G.H. Hardy, S. Chapman,  "A General View of the Theory of Summable Series".  QJM, 42 (1911) pp. 181-219
Lloyd L. Smail,  Some generalizations in the theory of summable divergent series.  ( PhD, 1913) 46 pp.
Lloyd L. Smail,  A General Method of Summation of... Annals of Mathematics II, 20, 3 (1919) pp. 149-154.


(2016-05-03)   The Stability Theorems
When is a series summable by convergence factors  stable ?

A summation method relying on convergence factors can be formulated as defining the sum...

 Come back later, we're
 still working on this one...


(2012-08-09)   Weierstrass Summation   (1842)
Analytic continuation  viewed as a summation method...

This applies not only to  formal power series  outside of their disk of convergence but also when each term of the series is an analytic function of  z  indexed by  n  (the  zeta series being the prime example of that).

 Come back later, we're
 still working on this one...


(2018-07-28)   Resummation with Meijer G-functions

Wikipedia :   Meijer G-function (1936)   |   Cornelis Simon Meijer (1904-1974)
 
Fast Summation of Divergent Series and Resurgent Transseries in Quantum Field Theories from Meijer-G Approximants by  Hector Mera,  Thomas G. Pedersen  &  Branislav K. Nikolic  (2018-02-16),


(2012-07-28)   Zeta Function Regularization   (1916)
A summation method which makes mincemeat of some  unstable  series.

This approach was pioneered by  G.H. Hardy  &  J.E. Littlewood.

To any  arithmetic function  f  is associated a  Dirichlet series  F(z) :

F(z)  =  Sn*  f (n) n-z

If  f  is subexponential and the real part of  z  is large enough,  then the series converges.  Otherwise we can extend it by analytic continuation  (as is normally done to define the zeta function itself)  to obtain the desired sum as the value of  F(0)  whenever it makes sense that way.

This is well-defined because of the  uniqueness of Analytic Continuation.  The linearity can be established by noticing that the map from a function to its Dirichlet series is a linear one and analytic continuation preserves linearity at every point where it makes sense,  including  z=0  supposedly.

Examples:

1  +  1  +  1  +  1  +  1  +  ...   =   - 1/2     =   z (0) 
1  +  2  +  3  +  4  +  5  +  ...   =   - 1/12   =   z (-1)

We've already noted the  unstability  of the first example.  The unstability of the second one results from the aforementioned linearity of this summation method.  Indeed, by adding or subtracting both sides of the above pair of equations, we obtain:

2  +  3  +  4  +  5  +  6  +  ...   =   - 7/12
0  +  1  +  2  +  3  +  4  +  ...   =   + 5/12

Wikipedia :   Dirichlet series  and  "abscissa of convergence"
Zeta function regularization   |   Heat-kernel regularization   |   1 + 1 + 1 + 1 +...   |   1 + 2 + 3 + 4 +...


 Gerard Michon (2016-04-27)   Wonders of Unstable Summations
Applying  linear  summation methods to unstable series.

In the  previous section,  we saw examples of series whose lack of stability could be demonstrated using the linearity of summation.  I'm not prepared to abandon the linearity requirement for summation methods, even if that means giving up the "obvious" stability property.  In this section, I'll show some of the consequences in more details.  For shock value.

Let's revisit the previous argument with the following notations.  In the first line, we explicitly invoke the assumption that a finite sum is equivalent to a series ending with infinitely many zeroes.

1  +  ... +  1  +  0  +  0  +  0  +  0  +  0  +  ...   =     m 
0  +  ... +  0  +  1  +  1  +  1  +  1  +  1  +  ...   =   A
0  +  ... +  0  +  1  +  2  +  3  +  4  +  5  +  ...   =   Bm  

In those last two cases,  the integer  m  denotes the number of zero terms the series starts with.  It matters.  By linearity  only,  we see that   Am  =  A0 - m

Also by linearity, the following recurrence holds:   Bm+1   =   Bm - Am

As we know  the values of  A0  and  B0 ,  we can solve this and obtain:

Am   =   -m - 1/2           and           Bm   =   m2/2 - 1/12

The above shows that  all  linear summation methods will give those surprising results,  as long as they agree with  zeta summation  for m=0.


 Gerard Michon (2018-05-26)   Decimated Series
Working out decimated unstable series using a few  stable  ones.

First,  a simple example using our favorite unstable serie and its alternating counterpart  (Grandi's series)  which is stable,  albeit divergent:

1  +  1  +  1  +  1  +  1  +  1  +  1  +  ...   =   -½
-  1  +  1  -  1  +  ...  +    (-1) +  ...   =    ½ 

By linearity only,  half the sum and half the difference of those two are:

1  +  0  +  1  +  0  +  1  +  0  +  1  +  ...   =   0  
0  +  1  +  0  +  1  +  0  +  1  +  0  +  ...   =   -½

We could subtract a finite sum from either one of those to obtain the sum of a  2-decimated  series,  starting with an odd or an even number of zeroes.  (This would result in a  seamless  formula,  which we're about to generalize.)

Decimated Geometric Series

Recall the basic facts about the  geometric series:

1  +  z  +  z +  z +  z +  ...  +  z +  ...   =   1 / (1-z)

This series is convergent for  |z| < 1.  It's divergent but  stable  everywhere else in the  complex  plane,  except at the pole of the right-hand-side  (z=1)  where the series is  unstable  of sum  -½  (obtained by zeta regularization).

We'll use that series when  z  is a primitive  k-th  root of unity to obtain,  by linearity alone,  the sum of the  decimated  series where only every k-th term subsists  (we demonstrate the technique  elsewhere  for convergent series).

Our preliminary example was for  k=2  (the only primitive square root of unity is  z = -1).  Let's do it again for  k=3,  with  z = w = exp (2pi/3)

w  =  1               1  +  w  +  w2   =   0

The first 3 lines below correspond to  z = 1, w and w2.  The other ones are just linear combinations thereof,  using the coefficients highlighted at right.

1 
1 
1 
 + 
 + 
 + 
1 
w 
w2
 + 
 + 
 + 
1 
w2
w 
 + 
 + 
 + 
1 
1 
1 
 + 
 + 
 + 
1 
w 
w2
 + 
 + 
 + 
1 
w2
w 
 + 
 + 
 + 
1 
1 
1 
 +  ...   =   -1 / 2
 +  ...   =   1 / (1-w)
 + ...   =   1 / (1-w2 )
 | 1 
1 
1 
 | 1 
w2
w 
 | 1 
w 
w2
3   +  0   +  0   +  3   +  0   +  0   +  3   +  ...   =   1 / 2
0   +  3   +  0   +  0   +  3   +  0   +  0   +  ...   =   -1 / 2
0   +  0   +  3   +  0   +  0   +  3   +  0   +  ...   =   -3 / 2

The reader is encouraged to work out the above left-hand sides  (mentally)  and  right-hand sides  (with pencil and paper).

Check that adding our three results gives just three times the top equation:

3  +  3  +  3  +  3  +  3  +  3  +  3  +  ...   =   -3 / 2

Let's repeat the above pattern for  k=4  with  w = exp (2pi/4)

w  =  1               1  +  w  +  w2  +  w3   =   0

1 
1 
1 
1 
 + 
 + 
 + 
 + 
1 
w 
w2
w3
 + 
 + 
 + 
 + 
1 
w2
 1 
w2
 + 
 + 
 + 
 + 
1 
w3
w2
w 
 + 
 + 
 + 
 + 
1 
1 
1 
1 
 + 
 + 
 + 
 + 
1 
w 
w2
w3
 + 
 + 
 + 
 + 
1 
w2
1 
w2
 +  ...   =   -1 / 2
 +  ...   =   1 / (1-w)
 + ...   =   1 / (1-w2 )
 + ...   =   1 / (1-w3 )
 | 1 
1 
1 
1 
 | 1 
w3
w2
w 
 | 1 
w2
1 
w2
 | 1 
w 
w2
w3
4   +  0   +  0   +  0   +  4   +  0   +  0   +  ...   =   1
0   +  4   +  0   +  0   +  0   +  4   +  0   +  ...   =   0
0   +  0   +  4   +  0   +  0   +  0   +  4   +  ...   =   -1
0   +  0   +  0   +  4   +  0   +  0   +  0   +  ...   =   -2

More generally,  the result of rank  m  consists of  k  times the sum  S0(k,m)  of the series formed by retaining the term of rank  m  and  every  k-th  term thereafter  (0 ≤ m < k).  It's obtained with  w = exp (2pi/k)  by assigning a coefficient  w-m q  to the equation of rank q.  That yields:

k  S0(k,m)     =     -1/2  +    k-1   w - m q / (1-w q )
S
q=1

When all is said and done  S0(k,k-1) is always equal to  -1/2.

S0(k,m)   =   1/2 - (m+1) / k       for   0 ≤ m < k

The validity of this formula extends to larger values of  m,  since subtracting one unit from the leading term of such a  k-decimated series clearly lowers the sum by one.  But that can also be construed as increasing  m  by k  units.  To summarize:

 m  leading zeroes   Period  k   
0 +  0 +  0 +  0 +  1 +  0 +  0 +  1 +  0 +  0 +  ...   =   ½ - (m+1) / k

From this,  we could work out the sum of any  ultimately periodic  series.

Decimated Power Series

The method and the notations just exemplified for the geometric series apply unchanged to the decimation  (0 ≤ m < k)  of any summable power series,  convergent or not  (HINT: multiply each of the above columns into its own coefficient).  Calling  f (z)  the undecimated sum,  we obtain:

Decimation Formula
[ f  is analytic on the unit circle except at 1.  f (0) ¹ 0 ]
    Sf (k,m)    =    
1
Vinculum
k
    k-1     exp (- 2pi mq/k )   f ( exp (2pi q/k) )  
S
q=0

Let's use that formula to decimate the series   1 + 2 + 3 + 4 + 5 + ...   knowing that   Sn* n zn-1 = f (z)   is  -1/12  if  z=1  and  1/(1-z) otherwise.

S1 m=0m=1m=2m=3m=4m=5m=6m=7m=8m=9
k=1 -1/12-13/12-37/12-73/12 -1/12 - m (m+1) / 2  
k=2 1/12-1/6-11/12-13/6-47/12 -37/61/12 - m/ 4  
k=3 1/121/12-1/4-11/12-23/12-13/4 1/12 - m (m-1) / 6  
k=4 1/241/61/24-1/3-23/24-11/6 1/24 - m (m-2) / 8  
k=5 -1/6011/6011/60-1/60-5/12-61/60 -1/60 - m (m-3) / 10  
k=6 -1/121/61/41/6-1/12-1/2
k=7 -13/8411/8423/8423/8411/84-13/84-7/12
k=8 -11/481/1213/481/313/481/12-11/48-2/3
k=9 -11/361/361/413/3613/361/41/36-11/36-3/4
10 -23/60-1/3013/6011/305/1211/3013/60-1/30-23/60-5/6
S1(k,m)   =   -k / 12  -  (m+1) (m+1-k) / 2k

The white entries  (for  m<k)  come directly from the  decimation formula.  Grey entries above the diagonal satisfy the recurrence  (for p=1):

Sp(k,m+k)   =   Sp(k,m)  -  (m+k)p

This establishes,  by induction,  the  unified formula  on the bottom line.

Of particular interest is the diagonal  (m = k-1)  where the sum of the decimated series is just  k  times the sum of the undecimated series  (-1/12).  Since the remaining terms are just the successive multiples of  k  in this case,  we may divide by  k  and retrieve a series which differs from the original one only by being  stretched  with  k-1  zero terms  before  each original term.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + ...
0 + 0 + 1 + 0 + 0 + 2 + 0 + 0 + 3 + ...

In our previous discussion of the constant series,  we had already remarked  that this type of padding always left the value  (-1/2)  of the sum unchanged.

To the uninitiated,  that property may look more obvious than it actually is:  Stretching  a series this way  often  doesn't change the sum,  but it  may...

To decimate the series of  squares   1 + 4 + 9 + 16 + 25 + ...   we use   Sn*  n2 zn-1 = f (z)   with  f (z)  =  (1+z)/(1-z)3  if  z¹1  and  f (1) = z(-2) = 0.

S2 m = 0m = 1m = 2m = 3m = 4m=5m=6m=7m=8m=9
k=1 0
k=2 00
k=3 -1/91/90
k=4 -1/401/40
k=5 -2/5-1/51/52/50
k=6 -5/9-4/904/95/90
k=7 -5/7-5/7-2/72/75/75/70
k=8 -7/8-1-5/805/817/80
k=9 -28/27-35/27-1-10/2710/27135/2728/270
10 -6/5-8/5-7/5-4/504/57/58/56/50
11 -15/11-21/11-20/11-14/11-5/115/1114/1120/1121/1115/11
12 -55/36-20/9-9/4-16/9-35/36035/3616/99/420/9
13 -22/13-33/13-35/13-30/13-20/13-7/137/1320/1330/1335/13
14 -13/7-20/7-22/7-20/7-15/7-8/708/715/720/7
S2(k,m)   =   -(m+1) (m+1-k/2) (m+1-k) / 3k

For  cubes,   Sn*  n3 zn-1  =  (1+4z+z2 )/(1-z)4  if  z¹1  or else  z(-3) = 1/120.

S3 m = 0m = 1m = 2m = 3m = 4m = 5m = 6
k = 1 1/120
k = 2 -7/1201/15
k = 3 -13/120-13/1209/40
k = 4 -7/240-7/15-7/2408/15
k = 5 29/120-91/120-91/12029/12025/24
k = 6 91/120-13/15-63/40-13/1591/1209/5
k = 7 1321/840-599/840-1919/840 -1919/840-599/8401321/840343/120
S3(k,m)   =   k3 / 120  -  (m+1)2 (m+1-k)2 / 4k

On the diagonal  m = k-1   the decimated sum is indeed  k3/120  as expected from the theorem heralded above,  which is the subject of the  next section.

Next,  we consider the series of  biquadrates  (or  fourth powers ).
Sn*  n4 zn-1   is  0  when  z=1  and   (1+11z+11z2+z3 )/(1-z)5   when  z¹1.

S4 m = 0m = 1m = 2m = 3m = 4m = 5m = 6m = 78
k = 1 0
k = 2 00
k = 3 1/3-1/30
k = 4 5/40-5/40
k = 5 74/2543/25-43/25-74/250
k = 6 17/316/30-16/3-17/30
k = 7 67/779/734/7-34/7-79/7-67/70
k = 8 119/820109/80-109/8-20-119/80
k = 9 196/9287/92794/9-94/9-27-287/9-196/90
S4(k,m)   =   - (m+1) (m+1-k) (m+1-k/2) [ (m+1)(m+1-k) - k2/3 ] / 5k

Sn*  n5 zn-1   is  -1/252  if  z=1  else   (1+26z+66z2+26z3+z4 )/(1-z)6

S5 m = 0m = 1m = 2m = 3m = 4m = 5
k = 1 -1/252 -1/252 - m2 (m+1)2 [2m(m+1)-1] / 12  
k = 2 31/252-8/63
k = 3 121/252121/252-27/28
k = 4 31/504248/6331/504-256/63
k = 5 -4537/126012347/126012347/1260-4537/1260-3125/252
k = 6 -3751/252968/63837/28968/63-3751/252-216/7
S5(k,m)   =   -k5 / 252  -  (m+1)2 (m+1-k)2 [ (m+1)(m+1-k) - k2 / 2 ] / 6k

Sn*  n6 zn-1   is  0  if  z=1  else   (1+57z+302z2+302z3+57z4+z5)/(1-z)7

S6 m = 0m = 1m = 2m = 3m = 4m = 5
k = 1 0- m (m+1) (2m+1) [ 3 m (m+1) (m(m+1)-1) + 1 ] / 42  
k = 2 00
k = 3 -7/37/30
k = 4 -61/4061/40
k = 5 -278/5-169/5169/5278/50
k = 6 -455/3-448/30448/3455/30
k = 7 -16955/49-20855/49-9194/499194/4920855/4916955/49
k = 8 -5587/8-976-5465/805465/8976
k = 9 -11596/9-17585/9-1701-6010/96010/91701
10 -11094/5-17792/5-17623/5-10816/5010816/5
S6(k,m)   =   -X (X ( X-k2 ) + k4/3) (m+1-k/2) / 7k     with  X = (m+1)(m+1-k)

Sn* n7 zn-1 = (1+120z+1191z2+2416z3+1191z4+120z5+z6)/(1-z)8  or  z(-7)=1/240

S7 m = 0m = 1m = 2m = 3m = 4
k = 1 1/2401/240 - m2(m+1)2 [ m(m+1) (3 m(m+1)-4) + 2 ] / 24  
k = 2 -127/2408/15
k = 3 -1093/240-1093/240729/80
k = 4 -127/480-1016/15-127/4801024/15
k = 5 23789/240-62851/240-62851/24023789/24015625/48
k = 6 138811/240-8744/15-92583/80-8744/15138811/240
S7(k,m)  =  k7/240 - X2 (X (X - 4k2/3) + 2k4/3) / 8k   with  X = (m+1)(m+1-k)

There's room for one more,  using again the shorthand   X  =  (m+1)(m+1-k)

S8 m = 0m = 1m = 2m = 3m = 4
k = 1 0 -m(m+1)(2m+1)[m(m+1) {5m(m+1)(m(m+1)-2)+9} -3] / 90 
k = 2 00
k = 3 809/27-809/270
k = 4 1385/40-1385/40
k = 5 9826/56047/5-6047/5-9826/50
k = 6 207913/27207104/270-207104/27-207913/27
k = 7 167107/7207679/792194/7-92194/7-207679/7
S8(k,m)   =   - X [ X {X(X-2k2)+9k4/5} - 3k6/5 ] (m+1-k/2) / 9k

Some values of the Zeta function       Numericana :   Decimated convergent power series


 Gerard Michon (2018-06-13 13:00 PDT)   Stretched Series  (leading  zero paddings)
Properly stretching a series doesn't change its sum...  usually.

 Come back later, we're
 still working on this one...

The sum of a  stretched harmonic series  depends on the stretching factor  k :

k k k k
0 +  0 +  1/1 +  0 +  0 +  1/2 +  0 +  0 +  1/3 +  0 +  0 +  1/4   ...   =    Log (2p/k)


 Gerard Michon (2019-07-15 16:30 PDT)   Invariants of a Series
Quantities unaffected by a standard groups of transformations.

In particular,  we may want to consider transformations which preserve the limits of all  convergent  series.  This includes the  acceleration methods  described here:  Shanks' transformRobertson's transform  and my own  harmonic acceleration  or its  generalization.

An infinite family of invariants,  indexed by  k,  is formed by the sums of the  k-decimations  of a given series.  Those form again a series whose invariants are second-order invariants of the original series,  starting with the sum of the new series.  This decimation procedure may be carried on with the new series,  and so forth...


 Gerard Michon (2018-05-20)   Divergent Fourier expansions
Harmonic analysis  is a source of  summable  divergent series.

The Fourier expansion of the  tangent function  is a summable divergent series.  So is the expansion of its derivative:

½ tan (x/2)   =   sin x   -   sin 2x   +   ...   +   (-1)n+1 sin nx   ...
¼ cos-2 (x/2)   =   cos x   -   2 cos 2x   +   ...   +   (-1)n+1 n cos nx   ...

Don't even think about trying the latter with  x = p.  However,  with  x = 0,  that's the sum of a series we're already familiar with.  With  x = p/2,  we get:

½   =   0 + 2 + 0 - 4 + 0 + 6 + 0 - 8 + 0 + 10 + 0 - 12 + 0 + 14 + 0 - 16 + ...

This looks superficially like twice the aforementioned  familiar  series and it is  (for nontrivial reasons,  since zeroes could be importanta priori).


 Gerard Michon (2019-08-09)   Asymptotic Solutions to Differential Equations
Definite differential properties of some divergent series.

Several methods exist to solve an  ordinary differential equation  as an asymptotic series about an hypothetical zero value of a nonzero parameter.  The  WKB method  is one,  but there are many others.  In particular,  Carl Bender  has pointed out that the usual way to solve a  perturbed Schrödinger equation  expresses the solution as a divergent asymptotic series.

Conversely,  if we know  a priori,  the solutions corresponding to the values  0  and  1  of a parameter,  such methods may yield a definite sum at point  1  for the ensuing asymptotic series about  0.  Convergent or not.


 Nicholas Mercator, FRS 
 (1620-1687) (2012-11-01)   Mercator Series   (1668)
The integral of the geometric series :

z  +  z2/2  +  z3/3  +  z4/4  +  ...  +  zn/n  +  ...   =   - Log(1-z)

This series belongs jointly to the history of the  invention of logarithms  and the  inception  of  Calculus.  Several authors discovered it independently:

  • Grégoire de Saint-Vincent (1584-1667)  knew about it before 1647.
  • Jan Hudde (1628-1704)  discovered it in 1656.
  • Nicholas Mercator ( Niklaus Kauffman, 1620-1687)  was the first to publish the series, in his  Logarithmotechnia  treatise  (1668).
  • Isaac Newton (1643-1727)  also discussed the series, which is now called either the  Mercator series  or the  Newton-Mercator series.

The  Mercator series  converges absolutely for  |z| < 1.
For  z = -1,  it's an alternating series that converges to  -ln 2,  as shown in 1650 by Pietro Mengoli  (1626-1686).  For  z = 1, we obtain the  harmonic series which was first shown to be  divergent  by  Nicolas Oresme  (c.1350).


 Gerard Michon (2018-07-01)   Decimating the Harmonic Series
Dealing with a  multivalued  analytic continuation.

To apply the  decimation formula  to the  harmonic series,  we use:

1 + z/2 + z2/3 + z3/4 + z4/5 + z5/6 + z6/7 + ...   =   f (z)

f (1)   is  x   (a constant we'll identify later).  For  z¹1,  we have  essentially :

f (z)   =   - Log (1-z) / z

That's treacherous because  Log  is a  multivalued  complex function  (defined modulo 2pi )  and we must use a single analytic branch covering  continuously  the  closed  unit disk,  with the possible exception of  z=1.  It turns out that all  computer algebra systems  (CAS)  make the choice we need by implementing their  Log  function with a  branch cut  (or cliff)  on the negative half of the real axis.  For the sake of enlightenment,  let's turn the auto-pilot off:

Consider a point   z ¹ 1   on the unit circle:   z = eiq   with   0 < q < 2p

1-z= 1 - cos q  -  i sin q = 2 sin2 q/2  -  2 i sin q/2 cos q/2
= -2i  sin q/2  eiq/2 = 2  sin q/2  ei (q-p)/2

In the above restricted range of  q,   sin q/2   is  positive.  Therefore:

Log (1-z)   =   ln (2 sin q/2)  +  i (q-p)/2  +  i 2pn     (for some integer  n)

With  n = 0,  we get an analytic expression of  f (z)  valid on the unit circle punctured at z=1,  consistent with  f (0) = 1.  The  decimation formula  yields:

1
Vinculum
k
   [   x   -    k-1     exp (- 2pi (m+1) q/k )   { ln (2 sin pq/k)  +  ip (q/k - ½) }   ]
S
q=1

Sums  of  Decimated  Harmonic  Series
S(k,m) m = 0m = 1m = 2m = 3
k = 1 x x - 1 x - 1 - 1/2 x - 1 - 1/2 - 1/3
k = 2 (x + ln 2)/2 (x - ln 2)/2 (x + ln 2)/2 - 1 (x - ln 2)/2 - 1/2
k = 3 x/3  +  (ln 3)/6
 +  p Ö3/18
x/3  +  (ln 3)/6
 -  p Ö3/18
(x - ln 3)/3 x/3  +  (ln 3)/6
+  p Ö3/18 - 1
k = 4 x/4  +  (ln 4)/8
 +  p/8
x/4 x/4  +  (ln 4)/8
 -  p/8
(x - ln 4)/4

CAS  may have a hard time simplifying the expressions given by the  decimation formula  in this case.  For example, the following equality:

S(7,6)   =   (x - ln 7)/7

is equivalent to a nontrivial trigonometric identity:

7   =   8 (1 + sin p/14) (1 + cos p/7) (1 - sin 3p/14)

Wikipedia :   Mercator series


 Gerard Michon (2018-07-12 22:30 PDT)   Transfinite Sum  of the Harmonic Series :
x   =   ln 2p   =   1.837877066409345483560659472811235...

The  harmonic series was first proved not to converge by  Oresme, around 1350.  Yet, that divergent series can be assigned a finite value canonically:

1  +  1/2  +  1/3  +  1/4  +  1/5  +  ...  +  1/n  +  ...   =   Log 2p

 Come back later, we're
 still working on this one...

Quick numerical verification,  with arbitrary precision :

One easy way to validate numerically the result of the  transfinite  summation of a divergent series is to express that series as a sum of other series which are either  convergent  series  (whose sums are computed numerically with or without acceleration procedures)  or  divergent series  of known sums.  (Conveniently,  the series of the k-th powers of the positive integers has zero sum when  k  is any  positive even integer.)

To deal with the above case of the  harmonic series,  we'll use  countable additivity  to form a series with only three nonzero terms.  Our starting point is provided by one of the formal series which define  Bernoulli numbers  (divided by  n2,  for good measure):

  1= 1  - 1  + 1  +  0  - n2  +  0  + ... Bnk-2  + ... 
Vinculum Vinculum Vinculum Vinculum Vinculum Vinculum
n (en-1) n22n12720 k!
Sn   1= p2  - x  - 1  +  0  + 0  +  0  + ...  0  + ... 
Vinculum Vinculum Vinculum Vinculum
n (en-1) 6224

As the series on the left-hand-side converges nicely,  its sum  S  is easily obtained numerically  (0.6843...)  to arbitrary precision,  which yields:

x   =   p2 / 3  -  1 / 12  -  2 S   =   1.8378770664...   =   Log 2p

Truthfully,  the above  verification  is actually how the transfinite sum of the harmonic function was first  discovered  (on 2018-07-12)  using a  12-place  decimal value  (quickly obtained on a  handheld calculator)  which  Simon Plouffe's  Inverter  identified immediately.  A memorable  magic moment !

At that time,  I wasn't entirely sure that the postulate of  countable additivity  was legitimate,  but that simple result gave me a great  boost of confidence...

Arguably,  x  is the correct value to assign to  z  at the simple pole 1.  The above goes  against  the popular guess that  z(1)  ought to be defined as the  Cauchy mean  of  z  about 1  (the constant term of the Laurent expansion):

lim  z(1-h) + z(1+h)     =   g   =   0.57721566490153286060651209...
Vinculum
h®0 2

Stirling's formula  (1730)
 
Resummation of the harmonic series  by  Yasiru89  (Physics Forums, 2008-10-26)
Does any method of summation work on the harmonic series?  by  Qiaochu Yuan  (MathOverflow, 2009-10-29)
Can the harmonic series be summed with a summability method?  by  Anders Kaseorg  (Quora, 2013-08-11)
Sum of the Harmonic Series?  by  Ayesha  (StackExchange, 2014-01-25)
A summability methods which sum the harmonic series  by  Renato Faraone  (StackExchange, 2017-09-23)
 
What makes natural logarithms so special? (1:14:53)  by  Grant Sanderson (3Blue1Brown, 2020-05-08).
 
Things get weird at infinity (16:20)  by  Andrew  (Zach Star, 2020-08-11).

 Gaston Darboux
Gaston Darboux
 

(2019-11-16)   Darboux's Summation Formula   (1876)
A generalization of the Euler-Maclaurin formula  (1735).

n
å
k = 0
  (a-z)k   [ y(n-k) (1)  f (k) (z)   -   y(n-k) (0)  f (k) (a) ]
=   - (a-z)n+1   ò  1
 
0
  y(t)   f (n+1) ( tz + [1-t]a )   dt

In this,  f  is an  analytic function  and  y  is any polynomial of degree  n.  (We may prove this by  induction  on  n,  using  integration by parts.)

If  y  =  yn  is the  n-th  element of an  Appell sequence,  this yields:

 Come back later, we're
 still working on this one...

Darboux's formula   }   Umbral calculus

 Shrinivasa Ramanujan
Srinivasa Ramanujan
 

(2012-07-29)   Ramanujan's Irregular Summation  (1913)

If I tell you this, you will at once   
point out to me the lunatic asylum.  

Second letter of  Ramanujan  to  G. H. Hardy  (1913)  

 Come back later, we're
 still working on this one...

Sum of the Harmonic Series?  (Cauchy principal value, by Anixx)   |   Ramanujan summation (Wikipedia)
Ramanujan's Summation (pdf) by  Eric Delabaere   (December 2001, reported by Vincent Puyhaubert)
Ramanujan summation  (Wikipedia)  applied by Alejandro   |   Quora answer  by  Anders Kaseorg.
 
Abel-Plana formula  (Plana 1820, Abel 1823)   |   Giovanni Antonio Amedeo Plana (1781-1864).   |   Niels Abel (1802-1829)
 
Videos :   Ramanujan Summation (8:45)  by  James Grime  (singingbanana, 2016-05-01).
Ramanujan Summation (11:39)  by  Steve Chow  (blackpenredpen, 2018-12-28).


(2012-07-31)   Summations of p-adic Integers
Any p-adic series whose  nth  term is a multiple of  n!  converges !

The following series is clearly convergent in p-adic integers for any  p :

1  +  1  +  2  +  6  +  24  +  120  +  ...  +  n!  +  ...

That's because the result of the sum modulo  pk  is not influenced at all by the terms beyond a certain index  m  (namely, the least integer whose factorial is a multiple of  p).  This is also true if the radix  (p)  is not prime.

The decadic sum is  ...4804323593105039052556442336528920420920314
The 2-adic sum is  ...101110010110111111000011111011111101000011010

The same remarks apply to the  Euler-Gompertz series:

-  1  +  2  -  6  +  24  -  120  +  ...  +  (-1)n n!  +  ...

That series converges in p-adic integers for any radix p  (prime or not)  and the sum is not invertible for some of them, which may be perceived as its finite "factors".  Those are the divisors of the following product:

2 2 . 5 . 13 . 37 . 463 .   ...   ( A064384 )


(2012-08-03)   Stieltjes Functions  &  Moments
   Thomas Stieltjes 
 (1856-1894)
Thomas Stieltjes

Well before the more general notion of  distributions  was devised  (in 1944, by my late teacher Laurent Schwartz)  the Dutch mathematician Thomas Stieltjes  considered  measures  as generalized derivatives of functions  of bounded variations of a real variable.  Such functions are differences of two monotonous bounded functions; they need not be differentiable or continuous.  (Stieltjes got his doctorate in Paris, under Hermite and Darboux.)

The moment...

 Come back later, we're
 still working on this one...

The poles are  mocking up the cut...

 Come back later, we're
 still working on this one...

Stieltjes transformation   |   Stieltjes series   |   Stieltjes moment problem   |   Thomas Stieltjes (1856-1894)
 
Videos of Carl Bender at the PI  (2011):  9:56   |   1:15:55


(2012-07-30)   Shanks Transformation   (1955  &  R.J. Schmidt 1941)
The transform of a sequence has the same limit but better convergence.

Motivation :

In a convergent sequence of the form   An  =  L + u vn   we may extract the limit  L  from 3 consecutive terms, by eliminating  u  and  v  as follows:

  • An-1   =   L + u vn-1
  • An     =   L + u vn
  • An+1  =   L + u vn+1

So,   v   =   ( An - L ) / ( An-1 - L )   =   ( An+1 - L ) / ( An - L )
Therefore,   ( An - L ) 2   =   ( An-1 - L )  ( An+1 - L )   which implies :

L   =   ( An-1 An+1 - An2 ) / ( An-1 + An+1 - 2 An )

Thus, the right-hand-side of that expression forms a sequence whose terms are expected to be close to the limit of  An  even when  An  is not of the special form quoted above.

This motivates the following introduction of a new sequence  Sn ,  which is defined for positive indices whenever the denominator doesn't vanish:

Shanks transform  Sn  of the sequence  An
 
Sn   =    
 
An-1 An+1 - An2
vinculum
An-1 + An+1 - 2 An

We observe that Shanks' transformation  commutes with translation :

( [An-1+k] [An+1+k] - [An+k] 2 ) / ( [An-1+k] + [An+1+k] - 2 [An+k] )
  =   k  +  ( An-1 An+1 - An2 ) / ( An-1 + An+1 - 2 An )    QED

Thus,  wlg,  we may focus on the analysis of sequences whose limit is zero.  (The difference between a convergent sequence and its limit is of this type.)

An Shanks Transform of  An
vn0
1 / n1 / 2n
 1 / np~  1 / (p+1)np
(-1)n / n(-1)n+1 / [ 4n (n2 - ½) ]
  (-1)n / np    ~  (-1)n+1 p / [ 4 np+2 ]  

The table shows that the convergence of a sequence that alternates above and below its limit is greatly accelerated by  Shanks' transformation  (the distance to the limit is essentially divided by the square of the index n).  Shanks's transformation is thus highly recommended for alternating series.

No such luck when the sequence approaches its limit from one side only.  The Shanks transform then offers only marginal improvement  (by dividing the distance to the limit by a constant factor, which is usually 2 or 3).  In that case, the approach described in the next section is preferred.

Seki Kowa (1642-1708)   |   Steffensen's method   |   Johan Frederik Steffensen  (1873-1961)
Aitken's delta-squared process (1926)   |   Alexander Aitken  (1895-1966)
Shanks transformation   |   Dan Shanks  (1917-1996)


(2012-07-30)   Richardson Extrapolation   (1911  &  Takebe 1722)
Accelerating the convergence of   An   =   L  +  k1 / n  +  k2 / n2 + ...

This  (very common)  pattern of convergence is the case where the above transformation of Shanks  has the poorest performance.  By optimizing for this pattern, we'll provide a convergence improvement in cases where the Shanks transformation does not deliver.

The method is similar, we eliminate 2 parameters between 3 equations:

  • An-1 (n-1)2   =   L (n-1)2  +  k1 (n-1)  +  k2
  • An       n 2      =   L    n 2     +  k1     n     +  k2
  • An+1 (n+1)2   =   L (n+1)2  +  k1 (n+1)  +  k2

Subtract twice the second equation from the sum of the other two:

An-1 (n-1)2  -  2 An n 2  +  An+1 (n+1)2     =     2 L

This motivates the definition of the  (order 2Richardson transformation:

Richardson transform  Rn  of the sequence  An
Rn   =     (n-1)2 An-1  -  2 n2 An  +  (n+1)2 An+1
vinculum
2

Richardson's transform  is a  linear map  that commutes with translation.

So, without loss of generality we can restrict the analysis of its performance to convergent sequences whose limit is zero  (consider such a sequence as the difference between some other sequence and its limit, if you must).

An   Richardson Transform of  An   ~
1 /0
  1 / n 2  0
  1 / n 3   1 / n(n2-1)   1 / n 3  
  1 / n 4  (3n2-1) / n2(n2-1)2   3 / n 4  
  1 / n 5  (6n4-3n2+1) / n3(n2-1)3   6 / n 5  
  1 / n 6  (10n6-5n4+4n2-1) / n4(n2-1)4  10 / n 6  
  1 / n 7  (15n8-5n6+10n4-5n2+1) / n5(n2-1)5  15 / n 7  
  (-1)n / n     2n (-1)n+1
  (-1)n / n 2   2 (-1)n+1

Thus, unlike the Shanks transformRichardson's transformation  is  absolutely catastrophic  when applied to the partial sums of an alternating series.  For a typical  nonalternating  series, it does a perfect job at the cancellation of the leading terms it's designed to handle and leaves the next order of magnitude virtually untouched.  However, the bad influence of higher-order error terms is significantly amplified  (possibly fatally so).

Caution!

Takebe (or "Tatebe")   =   Takebe Katahiro   =   Takebe Kenko (1664-1739)
"Our" Takebe was the younger brother of  Takebe Kataaki (1661-1716)  also a student of  Seki.

Wikipedia :   Richardson extrapolation   |   Lewis Fry Richardson  (1881-1953)


 Gerard Michon (2012-09-29)   On a new acceleration method   (Michon, 2012)
Accelerating the convergence of   An   =   L  +  k / (n-a)  +  ...

The transformations presented in the previous section are somewhat unsatisfactory because they involve explicitly the particular indexation of the sequence  (the value of n).  Clearly, if we tune a convergence acceleration to a truncated expansion of the shape presented here, the index n won't be involved because the presence of the parameter  a  makes the optimal result invariant by translation of the index.

Note that, if the correction terms of order  1/n3  and beyond are neglected, our new target is of the same magnitude as our previous one, with  k1 = k   and   k2 = ak.

Again, we eliminate 2 parameters between 3 equations:

  • An-1 (n - a - 1)   =   L (n - a - 1)  +  k
  • An    (n - a)         =   L (n - a)        +  k
  • An+1 (n - a + 1)   =   L (n - a + 1)  +  k

Subtract twice the second equation from the sum of the other two:

( An-1 + An+1 - 2 An ) (n - a)  +  ( An+1 - An-1 )   =   0

Let's also subtract the second equation from the third:

( An+1 - An ) (n - a)  +  An+1   =   L

Eliminating  (n - a)  between those two equations, we obtain:

L   =   [ 2 An-1 An+1 - An ( An-1 + An+1 ) ]  /  ( An-1 + An+1 - 2 An )

This motivates the following definition of a new sequence  Bn ,  which is valid for positive indices whenever the denominator doesn't vanish:

Transform  Bn  of the sequence  An
 
Bn   =    
 
2 An-1 An+1 - An ( An-1 + An+1 )
vinculum
An-1 + An+1 - 2 An

You may want to note that the  Shanks transform  of  An  is  (An+Bn)/2.

As this new transform commutes with translation  (the reader is encouraged to check that directly)  we may study its performance, without loss of generality,  for sequences whose limits are zero:

An   Bn =     Bn  ~  
1 /0
  1 / n 2   -1 / (3n2-1)   -1 / 3n 2  
  1 / n 3   -3n / (6n4-3n2+1)   -1 / 2n 3  
  1 / n 4  -(6n2+1) / (10n6-5n4+4n2-1)   -3 / 5n 4  
  (-1)n / n     -2n (-1)n / (2n2-1) - (-1)n / n
  (-1)n / n 2   -(2n2+1) (-1)n / (2n4-n2-1) - (-1)n / n2
  vn  -vn

Thus, the above transform is very effective when the leading error term is harmonic  (1/n).  For other types of convergence, the above table suggest using a linear mix of A and B for best acceleration,  as investigated next.


 Gerard Michon (2012-09-30)   Universal Convergence Acceleration  
Accelerating all typical analytical sequences.

Building on the above, let's introduce a parameter  u  and define:

A'n   =   [ (1-u) An + (1+u) Bn ] / 2

This way, the original sequence is obtained for  u = -1,  its Shanks transform for  u = 0  and the sequence  B  of the previous section for  u = 1.

 
    A'n   =    
 
(1+u) An-1 An+1  -  u An ( An-1 + An+1 )  -  (1-u) An2      
vinculum
An-1 + An+1 - 2 An
 
   =    
 
An-1 An+1  -  An2  -  u (An - An-1 ) (An+1 - An )      
vinculum
An-1 + An+1 - 2 An

Or, equivalently:

Parametrized transformation of the sequence  An
 
    A'n   =   An   +   (1+u)  
 
(An - An-1 ) (An+1 - An )      
vinculum
(An - An-1 ) - (An+1 - An )

The invariance by translation of this parametrized transform allows us to study it only for sequences whose limit is zero  (without loss of generality among convergent sequences).  Of course, we'll seek the value of  u  which provides the best acceleration of convergence.

To use the example already analyzed, if  An = L+k/n   then

A'n = L + (1-u) k/n

As we already know, the best value of  u  is indeed  +1  (which yields a constant sequence equal to the limit  L).  Here are a few other cases:

Optimal values of  u
An u   A'n  ~  
1 /10
  1 / n 2     1/2     -1 / 12n 4  
  1 / n 3   1/3   -1 / 6n 5  
  1 / n 4   1/4   -1 / 4n 6  
  1 / n 5   1/5   -1 / 3n 7  
  1 / n 6   1/6   -5 / 12n 8  
  1 / n p   1/p   (1-p) / 12n p+2  
  (-1)n / n   0   -(-1)n / 4n 3  

For the partial sums of alternating series, the Shanks transform  (u=0)  is optimal.  Otherwise, we can typically build an optimized sequence as:

An ,   A'n ,   A''n ,   A'''n ,   A''''n ,   A'''''n ,   ...

For this, we use a special sequence of different parameters determined by the expected way the sequence approaches its limit asymptotically.  Typically  (but not always)  the original sequence approaches its limit with a remainder roughly proportional to  1/n  and one order of magnitude is gained with each iteration using the sequence:

u0   =   1 ,   u1   =   1/2 ,   u2   =   1/3 ,   ...   un+1   =   1 / (1 + 1/un )

The computation is particularly easy to perform using a spreadsheet calculator.  We illustrate this by the following computation to 6 decimal places of the sum of the reciprocals of the squares, based on the first 7 terms in the series  (9 terms are given to show that the last two are useless).  Highlighted in blue are the  Shanks transforms  of the extreme diagonals.

z(2)   =   p2/ 6   =   1.644934066848226436472415...
nAn u0 = 1u1 = 1/2u2 = 1/3u3 = 1/4
11.000000 1.644704  
21.2500001.650000 1.644921 
31.3611111.6468251.644661 1.644934
41.4236161.6458331.6448111.644934 
51.4636111.6454291.6448681.6449341.644934
61.4913891.6452351.6448951.644934 
71.5117971.6451301.644909 1.644934
81.5274221.645069 1.644931 
91.539768 1.644909  

Although many terms of the basic sequence would be easy to compute in this didactic example, the method is meant to handle situations where this is not the case.  In theoretical physics  (quantum field theory)  and pure mathematics, we may only have a few terms available and only a fuzzy understanding of the behavior of the sequence whose limit has to be guessed as accurately as possible.

Incidentally, with standard limited-precision floating-point arithmetic, the relevant computations presented above will be very poorly approximated because we keep subtracting nearly-equal quantities.  As a rule of thumb, about half the available precision is wasted.  A 13-digit spreadsheet is barely good enough to reproduce the above 6½-digit table.  Extensions of it would be dominated by the glitches caused by limited precision.

Such pathological behavior is lessened by the approach described next.


(2012-10-02)   Accelerating a series by transforming its terms.
The series counterpart of the  parametrized  transform for sequences.

If the sequence  An  is the partial sum of the series of term  an ,  then we have  an = An - An-1   (for n≥1)  and the above boils down to:

 
    A'n   =   An   +   (1+u)  
 
an an+1      
vinculum
an - an+1

Subtracting from that value the counterpart for  A'n-1 ,  we obtain:

 
    a'n   =   an   +   (1+u)  
 
an an+1   -   (1+u)   an-1 an      
vinculum vinculum
an - an+1 an-1 - an

 Come back later, we're
 still working on this one...


(2012-09-30)   The next order...
An aborted attempt.

Let's target sequences of the form   An   =   L  +  k1 / (n-a)  +  k2 / (n-a)2

For the purpose of the following computation, we get rid of indices by considering four consecutive terms in the sequence  (A,B,C,D)  and introducing the quantity  x  that differs from  (n-a)  by some integer.  We seek an expression of the limit  L  as a function of  A,B,C,D  by eliminating the three quantities  x,  k1  and  k2  between the following four equations:

  •   A (x+0) 2   =   L (x+0) 2  +  k1 (x+0)  +  k2         [ 1]   [ 1]
  •   B (x+1) 2   =   L (x+1) 2  +  k1 (x+1)  +  k2         [-3]   [-2]
  •   C (x+2) 2   =   L (x+2) 2  +  k1 (x+2)  +  k2         [ 3]   [ 1]
  •   D (x+3) 2   =   L (x+3) 2  +  k1 (x+3)  +  k2         [-1]   [ 0]

The two columns of coefficients yield respectively these combinations:

( A - 3B + 3C - D ) x2  +  ( -6B + 12C -6D ) x  +  ( -3B + 12C - 9D)   =   0
( A - 2B + C ) x2  +  ( -4B + 4C ) x  +  ( -2B + 2C )   =   2L x

Eliminating  x  between those two  quadratic  equations yields:

|     A-3B+3C-D
A-2B+C
   -3B+12C-9D
-2B+2C
  |  2     =    

|     A-3B+3C-D
A-2B+C
   -6B+12C-6D
-2B+4C-2L
  |   .   |     -6B+12C-6D
-2B+4C-2L
   -3B+12C-9D
-2B+2C
  |

Unfortunately, this is now a quadratic equation in  L.


 James Stirling (the Venetian) 
 1692-1770 (2018-05-15)   Summation by Factorial Series
A forgotten method introduced by  James Stirling  in 1730.

 Come back later, we're
 still working on this one...

Summation of Divergent Power Series by Means of Factorial Series  by  Ernst Joachim Weniger  (2010-05-04).


(2017-06-24)   Padé Approximant
Simplest rational function with prescribed  truncated  Taylor expansion.

 Come back later, we're
 still working on this one...

Wikipedia   |   Weisstein   |   Henri Padé (1863-1953; ENS 1883)
The unreasonable effectiveness of Padé approximations  by Felix Goldberg   (MathOverflow, 2013-02-21)


(2017-08-08)   From Renormalization to Dimensional Regularization
The bag of tricks physicists use to make sense out of divergent series.

 Come back later, we're
 still working on this one...

Regularization & Renormalization of Gauge Fields  by  Gerard 't Hooft  &  Martinus Veltman  (1972).
 
Wikipedia :   Renormalization   |   Regularization   |   Dimensional regularization


(2019-11-21)   Physical meaning for the summation of divergent series
Why divergent series may be unavoidable in quantum physics.

In the tamest type of infinite series,  the  absolutely convergent  series,  all the familiar propertie of finite sums are preserved,  including commutativity and associativity.  This comfortable fact may be useful for numerical applications but it's simply roo restrictine for the way our quantum world seems to be constructed.

A certain historical order is essential to the way infinitely many probability amplitudes are combined, yet trajectories are not defined.  There's no such thing as a partial history.  Only the whole can be considered.  The whole sum need not be the limit of partial sums.

 Come back later, we're
 still working on this one...

The quantum substitute for logic


(2021-06-20)   Compensated  fractional derivations
They're defined in any  ring  or  power-associative algebra.

Fractional derivatives  are part of mathematical folklore.  They were first defined in 1695 by  Leibniz  in a letter to  Guillaume de l'Hospital.  They apply in any context where  fractional exponents  are defined.  They can be motivated by the following expression  (easily established by induction on k)  for the continuous linear operator  Dk  defined as k repetions of the ordinary derivation operator  D = D1  when k is a positive integer  (this expression is consistent with the convention that  D0  is the identity operator)  when it's applied to a monomial.  By linearity, that definition extends to all polynomials and, by continuity, to all analytic functions.

Dk xm   =   [ m! / (m-k)! ]  xm-k

Using the Gamma function, we obtain an equivalent expression which remains valid when k isn't an integer.

Dk xm   =   [ G(m+1) / G(m-k+1) ]  xm-k

That expression holds for  almost  all values of k, even complex ones  (exceptions arise because G is undefined for negative odd integers).  In this,  x  is understood to be a positive real number  (otherwise, raising x to a fractional power is problematic).  To remove that restriction,  we may consider instead the  compensated  operator which is defined in any power-associative algebra, including rings of matrices:

Jk xm   =   [ G(m+1) / G(m-k+1) ]  xm

Jk  doesn't change the degree of a polynomial, unless said degree is less than k.

 Come back later, we're
 still working on this one...

Gamma function   |   Fractional_calculus   |   Grünwald-Letnikov derivative (1867, 1868)
 
Half-Derivative of x (20:05)  by  Peyam R. Tabrizian  (Dr Peyam, 2017-11-17).
Fractional Derivative Definition (9:21)  by  Peyam R. Tabrizian  (Dr Peyam, 2018-08-01).

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