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Related Links (Outside this Site)Complex Variables, Contour Integration by Joceline Lega (1998).
Padé and Algebraic Approximants applied to the Quantum Anharmonic Oscillator (pdf)
by Christopher Orth, University of Minnesota, Morris (2005-12-12)
Assigning Values to Divergent Series (pdf) by David M. Bressoud (2006-03-29)
Effective Resummation Methods for an Implicit Resurgent Function by Eric Delabaere (2006)
The Everything Seminar: Sum Divergent Series by Matt Noonan 1 | 2 | 3
Adding, Multiplying and the Mellin Transform by Greg Muller (2007).
The n-category Café (discussing papers by Tom Leinster, in 2006 & 2007):
Euler characteristic of a category (2006-10-11)
Euler characteristic of a category as the sum of a divergent series (2007-07-09)
This Week's Finds in Mathematical Physics by John C. Baez : 124 | 125 | 126 (1998)
Euler's Proof that z(-1) = -1/12 (2003) | The Number 24 (2008)
Open Problems in Asymptotics Relevant to Sequence Transformations with Special Emphasis to the Summation of Divergent Stieltjes Series
by Ernst Joachim Weniger, Universität Regensburg (1995-03-15).
Euler-Maclaurin, zeta & analytic continuation by Terry Tao (2010-01-10).
Basic Resummation Theory by H. Kleinert and V. Schulte-Frohlinde.
Resummation, regularization, renormalization by Jose Javier Garcia Moreta
The Devil's Invention: Asymptotic, Superasymptotic and
Hyperasymptotic Series by John P. Boyd, University of Michigan (1999).
Mathsspiel : Strange case (2006)... Euler-Masheroni (2009)... To sum or not to sum... Definition(s)... Diverging appeal... Borel, Abel, Cesaro, Nörlund: I, II, III.
StackExchange: | Divergent Series (recent) | Apéry |
Google Plus: | Summation methods by Refurio Anachro (2014-10-18).
On the sum of all positive integers by John Baez (2013-06-06).
Divergence of perturbation theory: Steps towards a convergent series.
by Gerardo Enrique Oleaga Apadula & Sergio A. Pernice (1998)
[ On the applicability of Lebesgue's dominated convergence theorem ]
Steepest descent | Convergence of perturbation theory
by Hitoshi Murayama (221A Lecture notes, UC Berkeley, Fall 2006).
Formal Power Series
Introductory videos :
Adding past infinity &
("Minute Physics" 2011, by Henry Reich).
One minus one plus one minus one... by James Grime (in "Numberphile" 2013, by Brady Haran).
1+2+3+4+5+... = -1/12 (again) by Ed Copeland & Tony Padilla (in "Numberphile" 2014).
Ramanujan: Making sense of 1+2+3+4+5+... = -1/12 by Burkard Polster (Mathologer, 2016-04-22).
Math History: Infinite series by Norman J. Wildberger (UNSW, 2011-06-06).
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 x [ Sn an fn (x) ] = lim x [ Sn an gn (x) ]
(In infinitely many dimensions, linear forms could be discontinuous.)
Sn an = Sn an | n > = | y >
A 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 < s | is stable, we have: < s | y > = < s | Û | y >
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 beetween a series and its sum. Thus, we express the above as:
The second 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 expression doesn't either. Two generations before him, the great Euler had taken the opposite view, rather freely, with great success (much later, Ramanujan would do the same). Cauchy, however, 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 classifed into two broad groups:
Sometimes, those two types of approaches are known to be equivalent.
What the Hahn-Banach extension theorem has to say...
By definition, an absolutely regular summation method is a continous 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, it may change the sum of other convergent series and, thus, induce a non-regular summation method.
The earliest method of summation by convergence factors.
Euler introduced the following definition :
Arguably, this is the first use of the postulated continuity of summation. For example, Euler famously derived the following sum:
1 - 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[
1 - 2 z + 3 z2 - 4 z3 + ... + (-1)n+1 (n+1) z n + ... = 1 / (1+z)2
Abel's limit theorem (1826)
All Nörlund means are consistent with one another.
Following Neils E. Nörlund (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 :
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 Nörlund mean. If A'n tends to a limit S as n tends to infinity, then S is called the Nörlund-sum of the series an or, equivalently, the Nörlund-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).
Nörlund means (1919)
Niels E. Nörlund (1885-1981)
Consistent with Nörlund summation.
The question of the consistency of Nörlund and Hausdorff methods was raised by E. Ullirich and it was answered (in the affirmative) by W. Jurkat and A. Peyerimhoff, in 1954.
The summation method proposed by Abel is:
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 an = Sn ò0¥ an 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 an = ò0¥ ( Sn an 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 = zn ) 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 furmula 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). He evaluated it in half a dozen distinct consistent ways:
1 - 1 + 2 - 6 + ... + (-1)n n! + ... = ò0¥ ( 1 + t ) - 1 e-t dt
1 + 1 + 2 + 6 + 24 + ... + n! + ... = ò0¥ ( 1 - t ) - 1 e-t dt
On Various Definitions of the
Sum of Divergent Series. (Dissertation,
When is a series summable by convergence factors stable ?
A summation method relying on convergence factors can be formulated as defining the sum...
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).
An unstable approach pioneered by G.H. Hardy & J.E. Littlewood.
F(z) = Sn f (n) n-z
When f is subexponential, that series converges when the real part of z is large enough and can otherwise be extended by analytic continuation (like the zeta function) whereby we obtain the value of F(0), except when 0 is a pole of F. Examples:
1 + 1 + 1 + 1 +
1 + ...
= - 1/2
= z (0)
We've already noted the unstability of the first example. The unstability of the second one results from the linearity of summation. Indeed, by adding or subtracting both sides of the above pair of equations, we obtain:
2 + 3 + 4 + 5 + 6 + ...
= - 7/12
The linearity we just used could be established by noticing that the map from a function to its Dirichlet series is a linear one. If the analytic continuation is unique (it may not be) this implies the linearity of the map which sends a function to the analytic continuation of its Dirichlet series or to the value of that continuation at z=0.
Applying linear summation methods to unstable series.
In the previous section we gave examples of series whose lack of stability could be demonstrated using the linearity of summation. I, for one, am 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 explicitely invoke the assumption that a finite sum is equivalent to a series ending with infinitely many zeroes.
1 + ... + 1 +
0 + 0 + 0 + 0 +
0 + ...
In the two latter cases, the integer n denotes the number of zero terms the series starts with. By linearity only, we see immediately that An = A0 - n
Also by linearity, the following recurrence holds: Bn+1 = Bn - An
Solving this, using zeta regularization for A0 and B0 , we obtain:
An = -n - 1/2 and Bn = n2/2 - 1/12
The integral of the geometric series :
z + z2/2 + z3/3 + z4/4 + ... + zn/n + ... = - Log(1-z)
This series belongs to the history of the invention of logarithms. It was discovered independently by several authors:
The Mercator series converges absolutely for |z| < 1.
Wikipedia : Mercator series
Assigning the Euler-Masheroni constant g to the harmonic series.
If I tell you this, you will at once point out to me the lunatic asylum.
1 + 1/2 + 1/3 + 1/4 + 1/5 + ... + 1/n + ... = g
Sum of the Harmonic Series?
(Cauchy principal value, by Anixx)
Rammanujan summation (Wikipedia)
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 pk ). This is also true if the radix (p) is not prime.
The decadic sum is
The same remarks apply to the Euler-Gompertz series:
1 - 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 )
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 transform of a sequence has the same limit but better convergence.
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:
v = ( An - L )
/ ( An-1 - L )
= ( An+1 - L )
/ ( An - L )
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:
We observe that Shanks' transformation commutes with translation :
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).
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.
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:
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 2) Richardson transformation:
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).
Thus, unlike the Shanks transform, Richardson'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).
Accelerating the convergence of An = L + k / (n-a) + ...
The transformations presented in the previous section are somewhat unsatisfactory because they involve explicitely 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 will not 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:
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:
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:
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.
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.
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:
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.
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.
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:
Subtracting from that value the counterpart for A'n-1 , we obtain:
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:
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:
Unfortunately, this is now a quadratic equation in L.
Only zero is asymptotic to zero.
Asymptotic expansions may or may not be convergent.
The Stirling series is a divergent asymptotic series.
Simplest rational function with prescribed truncated Taylor expansion.
Henri Padé (1863-1953; ENS 1883)