14 1. The Complex Numbers

5. For which values of z does the sequence {zn} converge?

6. Prove that if a sequence {zn} converges, then it is bounded – that is, there is a

positive number M such that |zn| ≤ M for all n. Hint: Show that there is an

N such that |zn| ≤ |z| + 1 for all n ≥ N. This gives an upper bound for |zn|

for all but finitely many of the zn.

7. Prove that if {zn} is a sequence with lim zn = w, then lim zn = w and lim |zn| =

|w|.

8. For the sequence of the previous exercise, prove that lim 1/zn = 1/w provided

w = 0.

9. Prove the term test for divergence of a series. That is, if the sequence of terms

{zk} does not have limit 0, then the series

∑∞

k=0

zk diverges.

10. Does the series

∑∞

n=0

n/(3 + 2ni) converge?

11. Does the series

∑∞

n=1

n/(n3

+ 2i) converge?

12. For which values of z does the series

∑∞

n=0

1/(n2 + z2) converge?

13. Find the radius of convergence of the power series

∑∞

n=0

nzn.

14. Find the radius of convergence of the power series

∑∞

n=0

zn/3n.

15. Find the radius of convergence of the power series

∑∞

n=0

zn/(1

+

2n).

16. Find the radius of convergence of the power series

∑∞

n=0

(n!/nn)zn.

1.3. The Exponential Function

There are many real-valued functions of a real variable that have natural extensions

to complex-valued functions of a complex variable. In fact, this is true of all real

functions that have convergent power series expansions. If f(x) =

∑∞

n=0

an(x−x0)n

is a real power series which converges on an interval of radius R about x0, then the

complex power series f(z) =

∑∞

n=0

an(z − x0)n converges in the open disc of radius

R about x0 (Exercise 1.3.10) and serves to extend f to a function f(z) defined for

z in the disc DR(x0).

One of the most important examples of the use of this technique is provided

by the exponential function. We know from calculus that

ex

= exp x =

∞

n=0

xn

n!

,

where the series converges to

ex

on the entire real line. We saw in Example 1.2.12

that this same series, with x replaced by the complex variable z, converges abso-

lutely on the entire complex plane. Thus, we get an extension of ex to a function

ez defined on C as follows:

Definition 1.3.1. For each z ∈ C, we define exp(z) =

ez by

ez

=

∞

n=0

zn

n!

.