Sequences and Series 197so, by Gauss's test, the series converges absolutely if Re( c + 1 - a - n) > 1,
or Re(c - a - b) > 0.4.10 Sequences and Series of Functions
In this section we discuss briefly sequences of functions of the form
{un(z)}~=O' as well as the associated series L:;~=O un(z). In the next sec-
tion we consider, in particular, series of the type L:;~=O anzn, called power
series. A more detailed discussion of sequences and series of functions is
given in Chapter 8.
Definition 4.10 Let { un(z)} be an infinite sequence of single-valued func-
tions un(z) all having a nonempty common domain of definition D. At a
point z 0 E D the sequence is said to converge or diverge depending on
whether the numerical sequence {un(z 0 )} converges or diverges. The set
of points C C D, and only those, for which {Un ( z)} is convergent is called
the convergence set of the sequence. This set may be empty, may coincide
with the whole of D, or may be a nonempty proper subset of D, its shape
depending on the nature of the terms of the sequence.
Assuming that C -:f. 0, at each point of C the sequence has a unique finite
limit, so a single-valued function f(z) = limn-+oo un(z) is defined on C.
By the definition of limit of a numerical sequence this means: For every
E > 0, there exists an N,,z > 0 (i.e., depending in general on E and on the
particular point z E C) such that n > N,,z implies that
Jf(z) - Un(z)J < EOn certain subsets B of C (at least on any finite subset of C) it happens
that there exists a positive number N, (depending on E but not on z) such
that n > N, implies that Jf(z) - un(z)J < E. On such sets B we say that
{un(z)} converges uniformly to f(z), and writeun(z) =i f(z), z EB, as n---+ oo
If the convergence in C is not itself uniform, we call it pointwise
convergence, denotedun(z)---+ f(z), z EC, asn-+oo
In the ·theory of functions the concept of uniform convergence plays a
more important role than that of pointwise convergence because it is usually
uniform convergence that is required for relevant properties of the terms
(continuity, differentiability, etc.) to be inherited by the limit function.