Math 461, Fall 2010
Chapter 3
Sequences and Series
Many of the more important functions studied in real analysis, such as the exponential or
trigonometric functions, are most conveniently defined using power series. The familiar power
series for the sine and cosine go back at least to Newton in 1676; they were derived again by
de Moivre in 1698, by James Bernoulli in 1702, and were widely used by Euler in the 1730s.
Power series are, if anything, even more important for the study of complex functions. In the
1820s, Cauchy made considerable use of power series (  an z n ) in a complex variable z. In
particular, any real function having a power series representation automatically gives rise to a
complex function with a corresponding complex power series, which provides a natural method
for generalizing functions from the real to the complex case. In the 1840s, Weierstrass showed
how to build up the whole of complex function theory using power series methods.
In this chapter we shall develop some elementary properties of sequences and series of
complex numbers, mostly by direct analogy with the real case, and then specialize to a deeper
study of power series.
1. Sequences
For our purposes, sequences are required only as a stepping-stone towards series.
Definition. A sequence in C is a function from the natural numbers to C. We usually write the
function values with subscripts instead of function notation: zn = f (n). The entire sequence is
denoted with parentheses: ( z n ) .
Definition. Let z be a complex number. A sequence ( z n ) in C has the limit z provided that for
every   0 , there exists a corresponding integer N such that for all n > N, | z n  z |   . If a
sequence has a complex limit z, it is said to be convergent, and we write lim z n  z . If there is
n 
no complex number z such that ( z n ) converges to z, the sequence is divergent.
The problem of finding limits of complex sequences can be reduced directly to the real case.
Theorem 1. Let ( z n )  ( xn  iy n ) be a sequence in C, where ( x n ) and ( y n ) are sequences in R.
Let z = x + iy, where x, y  R . Then lim z n  z iff lim x n  x and lim y n  y .
n 
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n
n 
Definition. Let ( z n ) be a sequence in C. We say ( z n ) is a Cauchy sequence provided the terms
of the sequence become arbitrarily close together as we move out into the tail of the sequence:
for all ε > 0, there exists an integer N such that for all m, n > N, | z m  z n |  .
The following is a direct consequence of the General Principle of Convergence for real
sequences (Cauchy sequences in R are convergent).
Theorem 2 (General Principle of Convergence). Let ( z n ) be a sequence in C. The following
are equivalent.
a) ( z n ) converges to some complex number z.
b) ( z n ) is a Cauchy sequence.
Exercise 3. Determine whether the following sequences converge, and find the limits of those
which are convergent.
a) z n  (1  i ) n
b) z n  (1  i ) n / n
c) z n  (1  i ) n / n!
d) z n  (1  i )  n
e) z n  n  (1  i)  n
f)
z n  n!(1  i )  n
Exercise 4. For which values of z do each of the following sequences converge?
a) z n  z n
zn
n
c) z n  n!z n
b) z n 
d) z n 
zn
n!
e) z n 
zn
, where k is a positive integer constant
nk
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2. Series
Definition. Given a sequence ( z n ) n 0 we can form a related sequence of partial sums defined by
n
s n   z r . We say that the series
r 0

z
r 0
r
converges to s (where s is a complex number) provided

the sequence of partial sums ( s n ) converges to s:
z
r 0
r
n
= lim s n  lim
n 
n 
z
r 0
r
. A series in C
which converges to some s in C is called convergent. If there is no such s, the series is divergent.

If
z
r 0
r
converges to s we can also write z 0  z1  z 2    s .
Any question about the convergence of a series can be turned into one about the sequence
( s n ) of partial sums. For example, using the General Principle of Convergence, we have the
following.

Theorem 5. Let
z
r 0
r
be a series in C. The following are equivalent.

a)
z
r 0
r
converges to some s in C.
b) The partial sums of the series are a Cauchy sequence: For all ε > 0, there exists an
integer N such that for all n, m  N , if n > m > N then |
n
z
r  m 1
r
| .
 3  4i 
Exercise 6. Let z n  
 . Use the previous theorem to determine whether or not the series
 6 
n

z
r 0
r
converges.
As for sequences, the summation of a complex series may be reduced to the equivalent real
series.
 z converges iff both of the real
series  x and  y converge, in which case we have  z   x  i y .
Theorem 7. Let z r  xr  iy r where xr , yr  R . Then
r
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r
r
r
r
r
Exercise 8. Let z n 
in
. Use the previous theorem to determine whether or not the series
n2

z
r 1
r
converges.
As in the real case, we can add convergent series and we can multiply them by constants.
 b be convergent series in C.
 (a  b )   a   b
 (ca )  c a
Theorem 9. Let
a)
b)
r
a
r
and
r
r
r
r
Let c  C .
r
r
When verifying that a complex series converges without computing its actual sum, it is often
better to concentrate on the modulus of the nth term rather than the real and imaginary parts.
Notice that | z r | is a real number, so all our knowledge about convergence of series in R can be

brought to bear on the question of whether or not
| z
r 0
r
| is convergent.

Definition. A complex series
z
r 0
r
is said to be absolutely convergent provided the series

| z
r 0
r
| is convergent. If a series is convergent but is not absolutely convergent, we say the
series is conditionally convergent.
Theorem 10. Every absolutely convergent series in C is convergent.

Exercise 11. Use absolute convergence to show that
Theorem 12 (Comparison Test). Let
a
r
and
ir
is convergent.

2
r 1 r
b
r
be complex series with
a
r
absolutely
convergent. If there exists an integer N > 0 and a real number K > 0 such that | br |  K | ar | for
all r > N, then
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b
r
is absolutely convergent, and hence convergent.
Theorem 13 (Ratio Test). Let
a
r
be a complex series with non-zero terms such that
| ar |
  for some real number  .
r  | a
r 1 |
lim
a
If  > 1, the series  a
If  = 1, the series  a
a) If  < 1, the series
r
is absolutely convergent.
b)
r
is divergent.
r
may be convergent or divergent.
c)
Exercise 14. For each series, determine whether or not the series is convergent.

i
a)   
r 0  2 
b) 1 

r
z2 z4 z6


  where z is an arbitrary complex number
2! 4! 6!
r
 z 
c)  
 where z is an arbitrary complex number, z ≠ 1
r 0  z  1 
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