1549901369-Elements_of_Real_Analysis__Denlinger_

512 Chapter 8 • Infinite Series of Real Numbers

(b) f has derivatives of all orders at every x in the interior of I.
00
(c) For any x in the interior of I , J fdx = L k ~
1
(x-c)k+I +C, where
k=O
C is an arbitrary constant.
(d) Vk EN, the kth derivative off at c is f(k)(c) = k! ak.

Proof. Parts (a)- (c) follow from Theorem 8.6.14. To prove (d), note that
J(^0 l(x) = a 0 +a 1 (x - c) + a2(x -c)^2 + a3(x - c)^3 + a4(x - c)^4 + as(x - c)^5 + · · ·
f (1) ( x) = a1+2a2 ( x - c) +3a3 ( x-c )^2 +4a4 ( x - c )^3 +5a 5 ( x-c )^4 +6a6 ( x-c )^5 + · · ·
f(^2 ) (x) = 2a 2 + 2 · 3a 3 (x-c) +3 ·4a 4 (x-c)^2 +4 · 5a 5 (x-c)^3 + 5 · 6as(x-c)^4 + · · ·
f(^3 l(x) = 2 · 3a 3 + 2 · 3 · 4a4(x - c) + 3 · 4 · 5a 5 (x - c)^2 + 4 · 5 · 6as(x - c)^3 + · · ·

When we substitute x = c into the above equations we get

j(o)(c) = ao = O! ao

f(ll(c) = a 1 = 1! a1

f(^2 l(c) = 2a2 = 2! a2

f (^3 l(x) = 2 · 3a3 = 3! a3

Corollary 8.6.16 (Uniqueness of Power Series Representation) If a

function f is representable as a power series with interval of convergence I ,

then that power series is unique. In fact,

00 f(k)( )

f(x) = L T(x - c)k

k=O

for every x in the interior of I.

Proof. Immediate consequence of Corollary 8.6.15. •

Definition 8.6.17 The series given in Corollary 8.6.16 is called the Taylor

series of f about c. In case c = 0, the series

00 f(k)( )

f(x)=L:~xk

k=O

is called the Maclaurin series of f.

Corollary 8.6.16 says that if a function has a power series representation

valid in an interval centered at c, it must be the Taylor series of f about c;

there are no other power series representations of f valid in this interval. We

shall now explore some implications of these results.