8.6 Power Series 50900
Definition 8.6.10 Given a power series L ak(x - c)k, the power series ob-
k=O 00
tained from it by differentiating it term-by-term, L kak(x - cl-^1 , is called
k=O
its derived series.Theorem 8.6.11 A power series and its derived series have the same radius
of convergence.
00
Proof. Consider a power series Lak(x - c)k. Note that its derived se-
k=O
1 00
ries can be written as --L kak(x - c)k. By Theorem 8.6.8, the radius of
x-c k=O
convergence of this series isk-+oo lim ~ = k-+oo lim ({/k ~)·
Recall from Example 2.3.8 that lim {/k = l. Thus, using Exercise 2.9.9,
k-+ook-+oo lim ~ = ( k-+oo lim {/k) ( k-+oo lim ~) = k-+oo lim ~'
which establishes the desired result. •00
Corollary 8.6.12 A power series L ak(x - c)k and its term-by-term inte-
k=O
00
grated series L kak
1(x - c)k+l have the same radius of convergence.
k=O +Proof. This is an immediate consequence of Theorem 8.6.11. •CAUTION: Theorem 8.6.11 does not say that the interval of convergence
of a power series is the same as that of its derived series. The two series may
behave differently at the endpoints of the interval. The following example should
make that clear.
00 k 00 k-1
Example 8.6.13 The power series L ~ 2 and its derived series LT h ave
k=l k=l
the same radius of convergence, p = l. However, the interval of convergence of
the given series is [-1, l] while the interval of convergence of its derived series
is [-1, 1).