Mathematics of Physics and Engineering

(^206) Power Series

4.3 Power Series and Analytic Functions

4.3.1 Series of Complex Numbers

A series of complex numbers ck, k > 0 is the infinite sum Co + c\ + c-i +... =
^fc>0Cfc. The sum ]Cfc=icfc' n ^ 1, is called the n-th partial sum of
the series. The series is called convergent if the sequence of its partial
sums converges, that is, if there exists a complex number C such that
linin^oo \C — X^fe=oCfcl = 0- The series is called absolutely convergent
if the series ^fc>olcfcl converges. A series that converges but does not
converge absolutely is said to converge conditionally. Note that the
sequence Yjk=i \ck\: n > 1, of the partial sums of the series ^2k>0\ck\
is non-decreasing and therefore converges if and only if Y^k=i lcfcl — C
for some number C independent of n. As a result, we often indicate the
convergence of Ylk>o \c^\ bY writing £fe> 0 lcfcl < °°.

EXERCISE 4.3.1.C Show that (a) the condition limn^oo \cn\ —> 0 is neces-

sary but not sufficient for convergence of the series ^Zk>0 ck; (b) absolute

convergence implies convergence, but not conversely.

Recall that, for a sequence of real numbers an, n>0, the upper limit

is denned by

limsupan= lim supofc,

n n-*°°k>n

where sup means the least upper bound. Since the sequence An =
supfc>n ak, n > 0, is non-increasing, the upper limit either exists or is equal
to +oo. For a convergent sequence, the upper limit is equal to the limit of
the sequence. Similarly, the lower limit liminf„a„ = linin^oo inffc>nafc,
where inf is the greatest lower bound, is a limit of a non-decreasing se-
quence; this limit is either —oo or a finite number, and, for a convergent
sequence, is equal to the limit of the sequence.

EXERCISE 4.3.2. Verify that, for every sequence {an, n > 0} of real
numbers, (a) the sequence {An, n > 0}, defined by An = supfc>nafc is non-
increasing, that is, An+i < An for all n > 0; (b) the sequence {Bn, n > 0},
defined by Bn = inf k>n ^fc is non-decreasing, that is, Bn+\ > Bn for all
n > 0. Hint. Use the following argument: if you remove a number from a finite
collection, then the largest of the remaining numbers will be as large as or smaller
than the largest number in the original collection; the smallest number will be as
small as or larger.