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!
.
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