1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

4. 1 • SEQUENCES AND SERIES 127

stipulates that convergent sequences are bounded. The same result holds for
complex sequences.

As with the real numbers, we also have the following definition.

Definition 4.3: Cauchy sequence

The sequence {z,.} is a Cauchy sequence if for every e > 0 there is a positive

integer N, such that ifn, m > N,, then lzn -zml <£,or, equivalently, Zn-Zm E

D, (O).

The following theorem should now come as no surprise.

One of the most important notions in analysis (real or complex) is a theory
t hat allows us to add up infinitely many terms. To make sense of such an idea we
begin with a sequence {z .. }, and form a new sequence {Sn}, called the seque nce
of partial sums, as follows.

81 = z1,

S2 = z 1 + z2,

83 = Z 1 + Z2 + Z3,

n

S,. = z1 +z2 + · · · +zn = 2:::;zk,

k=l