Nowadays, we call Taylor's Theorem
several variants of the following expansion of a smooth function
a regular point a, in terms of a polynomial whose coefficients are
determined by the successive derivatives of the function at that point:
f (a + x) =
f (a) +
f ' (a) x +
f '' (a) x/2 + ... +
f (n) (a) xn/ n! + ...
The variants of Taylor's theorem differ by the distinct explicit expressions that
can be substituted for the trailing ellipsis, which stands for the difference
between the value of the function and that of Taylor's polynomial.
Taylor published two versions of his theorem in 1715.
In a letter to his friend
Machin (1680-1751) dated July 26, 1712, Taylor gave Machin credit for the idea.
Several variants or precursors of the theorem had also been discovered independently by
James Gregory (1638-1675),
Isaac Newton (1643-1727),
Gottfried Leibniz (1646-1716),
Abraham de Moivre
A Taylor expansion about the origin (a = 0)
is often called a Taylor-Maclaurin expansion, in honor of
(1698-1746) who focused on that special case in 1742.
(2015-04-19) Basing calculus on Taylor's expansions (1772)
Lagrange's strict algebraic interpretation
of differential calculus.
Taylor's theorem was brought to great prominence in 1772 by
Joseph-Louis Lagrange (1736-1813)
who declared it the basis for differential calculus
(he made this part of his own lectures at Polytechnique in 1797).
The mathematical concepts behind differentiation and/or integration are so
pervasive that they can be introduced or discussed outside of the historical context
which originally gave birth to them, one century before Lagrange.
The starting point of Lagrange is an exact expression, valid for any polynomial f
of degree n or less, in any commutative ring :
f (a + x) =
f (a) +
D1 f (a) x +
D2 f (a) x2
+ ... +
Dn f (a) xn
In the ordinary interpretation of Calculus [over any field
of characteristic zero] the following relation holds, for any polynomial f :
D0 f (a) =
Dk f (a) =
f (k) (a) / k!
However, the above expansion remains true in very general circumstances
when neither the reciprocal of k! nor higher-order derivatives need be defined.
Lagrange's general definition of Dk f (a) is strictly based on the
Dk f is always a polynomial of degree n-k.
"Théorie des fonctions analytiques contenant les principes du calcul différentiel,
dégagés de toute considération d'infiniment petits ou d'évanouissants,
de limites ou de fluxions et réduits à l'analyse algébrique des quantités finies"
by Joseph-Louis Lagrange (1797)
Journal de l'École polytechnique, 9, III, 52, p. 49
(2008-12-23) Radius of Convergence of a Complex Power Series
A complex power series converges inside
a disk and diverges outside of it (the situation at different points of
the boundary circle may vary).
Brian Keiffer (Yahoo!
Formal properties of exp series.
Defining exp (x) = Sn xn/n!
and e = exp (1) prove that exp (x) = e x
In their open disk of convergence (i.e., circular boundary excluded)
power series are absolutely convergent series.
So, in that domain, the sum of the series is unchanged by modifying the
order of the terms (commutativity) and/or grouping them
This allows us to establish directly the following fundamental property
exp (x) exp (y) = exp (x+y)
Such manipulations are disallowed for convergent series that are not
absolutely convergent (which is to say that the series consisting of
the absolute values of the terms diverges).
Rearranging the terms of any such real series can make it converge
to any arbitrary limit !
exp (x) exp (y)
n = 0
n = 0
= exp (x+y)
This lemma shows immediately that exp (-x) = (exp x)-1.
Then, by induction on the absolute value of the integer n, we can establish that:
exp (n x) = (exp x) n
With m = n and y = n x, this gives
exp (y) = (exp y/m)m . So :
exp (y / m) = (exp y) 1/m
Chaining those two results, we obtain, for any rational q = n/m
exp (q y) = (exp y) q
By continuity, the result holds for any real q = x. In particular, with y = 1:
exp (x) = (exp 1) x
= e x
(2008-12-23) Analytic Continuation (Weierstrass, 1842)
Power series that coincide wherever their disks of convergence overlap.
Loosely speaking, analytic continuations
can make sense of divergent series in a consistent way.
Consider, for example, the classic summation formula for the
geometric series, which converges when
|z| < 1 :
1 + z +
z4 + ... +
zn + ...
= 1 / (1-z)
The right-hand-side always makes sense, unless z = 1.
It's thus tempting to equate it formally
to the left-hand-side, even when the latter diverges!
This viewpoint has been shown to be consistent.
It makes perfect sense of the following "sums" of
divergent series which may otherwise look like monstrosities
(respectively obtained for z = -1, 2, 3) :