Complex Analysis - Maths KU

6.2 Taylor Series 313
45. Consider the power series f(z) = �∞ k=0 akz k where
⎧
⎨ 2
ak =
⎩
k , k =0, 2, 4,...
1
k =1, 3, 5,... .
7k (a) Show that neither lim
n→∞ |an+1/an| nor limn→∞ |an|1/n exist.
(b) Find the radius of convergence of each power series:
∞�
f1(z) = 2 2k z 2k
and
∞�
f2(z) =
k=0
k=0
1
7 2k+1 z2k+1 .
(c) Verify that f(z) =f1(z) +f2(z). Discuss: How can the radius of convergence
R for the original power series be found from the foregoing observation?
What is R?
46. Proceed as in Problem 45 to find the radius of convergence R for the power
series
1+3z + z 2 +27z 3 + z 4 + 243z 5 + z 6 + ··· .
47. In this problem you are guided through the proof of the proposition:
If the series �∞ k=1 zk converges absolutely, then the series converges.
6.2Taylor Series
Proof We begin with the hypothesis that � ∞
then � ∞
k=1 | ak| ≤ � ∞
k=1
� a 2 k + b 2 k = � ∞
k=1 |zk|.
k=1 |zk| converges. If zk = ak+ibk,
(a) First, explain why the foregoing inequality is true. Second, explain why
this inequality shows that the series �∞ k=1 | ak| converges.
(b) Explain how your reasoning in part (a) also shows that �∞ k=1 | bk|
converges.
(c) Explain how parts (a) and (b) show that � ∞
k=1 zk converges.
The correspondence 6.2 between a complex number z within the circle of convergence and
the number to which the series � ∞
k=0 ak(z − z0) k converges is single-valued.In this sense,
a power series defines or represents a function f; for a specified z within the circle of
convergence, the number L to which the power series converges is defined to be the value of
f at z, that is, f(z) =L.In this section we present some important facts about the nature
of this function f.
In the preceding section we saw that every power series has a radius of convergence
R.Throughout the discussion in this section we will assume that a power series
� ∞
k=0 ak(z − z0) k has either a positive or an infinite radius R of convergence.
Differentiation and Integration of Power Series The three
theorems that follow indicate a function f that is defined by a power series is
continuous, differentiable, and integrable within its circle of convergence.