where c_{n} = ∑_{k=0}^{n}a_{k}b_{n−k}. Furthermore,∑_{n=0}^{∞}c_{n}converges absolutely.
Proof: It only remains to verify the last series converges absolutely. Letting p_{nk} equal
1 if k ≤ n and 0 if k > n. Then by Theorem 10.5.4 on Page 684
With this theorem it is possible to consider the question raised in Example 11.6.3 on
Page 760 about the existence of the power series for tanx. This question is clearly
included in the more general question of when
( ∞∑ ) −1
an(x − a)n
n=0
has a power series.
Lemma 11.8.5Let f
(x)
= ∑_{n=0}^{∞}a_{n}
(x − a)
^{n}, a power series having radius ofconvergence r > 0. Suppose also that f
(a)
= 1. Then there exists r_{1}> 0 and
{bn}
suchthat for all
|x− a|
< r_{1},
∑∞
--1--= bn (x − a)n .
f (x) n=0
Proof: By continuity, there exists r_{1}> 0 such that if
( )
∑∞ ∑∞ n ∑∞ ∑∞ n ∑∞ ∞∑ n p
|bnp||x− a| ≤ Bnp |x− a| = |cn||x− a| < ∞
p=0n=0 p=0n=0 p=0 n=0
by 11.10 and the formula for the sum of a geometric series. Since the series of 11.11
converges absolutely, Theorem 10.5.4 on Page 684 implies the series in 11.11
equals
∞ ( ∞ )
∑ ∑ bnp (x − a)n
n=0 p=0
and so, letting ∑_{p=0}^{∞}b_{np}≡ b_{n}, this proves the lemma. ■
With this lemma, the following theorem is easy to obtain.
Theorem 11.8.6Let f
(x)
= ∑_{n=0}^{∞}a_{n}
(x− a)
^{n}, a powerseries having radiusof convergence r > 0. Suppose also that f
(a)
≠0. Then there exists r_{1}> 0 and
{b}
n
suchthat for all
|x− a|
< r_{1},
1 ∑∞ n
f-(x) = bn (x − a) .
n=0
Proof: Let g
(x)
≡ f
(x)
∕f
(a)
so that g
(x)
satisfies the conditions of the above
lemma. Then by that lemma, there exists r_{1}> 0 and a sequence,
{bn}
such
that
∑∞
f (a)= bn (x − a)n
f (x) n=0
for all
|x− a|
< r_{1}. Then
∑∞
-1--= ^bn (x − a)n
f (x) n=0
where
^bn
= b_{n}∕f
(a)
. ■
There is a very interesting question related to r_{1} in this theorem. Consider
f
(x )
= 1 + x^{2}. In this case r = ∞ but the power series for 1∕f
(x )
converges
only if
|x|
< 1. What happens is this, 1∕f
(x)
will have a power series that will
converge for
|x− a|
< r_{1} where r_{1} is the distance between a and the nearest
singularity or zero of f
(x )
in the complex plane. In the case of f
(x)
= 1 + x^{2} this
function has a zero at x = ±i. This is just another instance of why the natural
setting for the study of power series is the complex plane. To read more on
power series, you should see the book by Apostol [3] or any text on complex
variable. The best way to understand power series is to use methods of complex
analysis.