8.7 Analytic Functions 519- Use Abel's theorem to prove Abel's claim that if the Cauchy product of
two convergent series converges, its sum must be the product of their
sums. (See notes after Theorem 8.4.4.) - 7 Analytic Functions
In the concluding pages of Section 8.6, we found that if a function f is rep-
resentable as a power series about a real number c then f must be infinitely
differentiable at c, and the kt/' coefficient in the power series is expressible in
terms of the kth derivative off at c. We also saw that there can be only one such
power series- the Taylor series off about c, as specified in Definition 8.6.17.
In this section we focus on the converse question: Given a function f that is
infinitely differentiable at a real number c, how can we tell if f is representable
as a power series in some neighborhood of c? We begin with a definition.
Definition 8. 7 .1 A function f is said to be analytic at c if there is a power
series about c that represents f in some neighborhood of c. Equivalently, f is
analytic at c if the Taylor series off about c converges to f(x) for all x in some
neighborhood of c.
So, given a function f that is infinitely differentiable at c, we want to
determine whether f is analytic at c. The fundamental tool used in answering
this question is Taylor's theorem, which was proved and discussed in Section
6.5. Let us take another look.
Definition 8. 7 .2 Suppose f is infinitely differentiable at c. Then, \In E N, the
nth Taylor polynomial for f about c is
n, J(k)(c) k
Tn(x) = .L:-k
1
-(x -c) ,
k=O(26)and the nth Taylor remainder for f about c is
Rn(x) = f (x) - Tn(x). (27)
Thus, Vx EI,
f (x) = Tn(x) + Rn(x). (28)To say that f has a power series representation in a neighborhood of c is
to say that for all x in this neighborhood, lim Tn(x) = f(x). Equivalently,
n->oo
lim Rn(x) = 0. Taylor's theorem is the principal tool we use in showing that
n->oo
this limit is O; it enables us to get a handle on Rn(x).
Theorem 8. 7.3 (Taylor's Theorem, with Various Forms of the Re-
mainder) Suppose f is n times differentiable on an open interval containing