sequence and series Functions

, Chapter 9




Sequences and Series of
Functions



In this chapter, we define and study the convergence of sequences and series of
functions. There are many different ways to define the convergence of a sequence
of functions, and different definitions lead to inequivalent types of convergence. We
consider here two basic types: pointwise and uniform convergence.


9.1. Pointwise convergence
Pointwise convergence defines the convergence of functions in terms of the conver-
gence of their values at each point of their domain.
Definition 9.1. Suppose that (fn ) is a sequence of functions fn : A → R and
f : A → R. Then fn → f pointwise on A if fn (x) → f (x) as n → ∞ for every
x ∈ A.

We say that the sequence (fn ) converges pointwise if it converges pointwise to
some function f , in which case
f (x) = lim fn (x).
n→∞

Pointwise convergence is, perhaps, the most obvious way to define the convergence
of functions, and it is one of the most important. Nevertheless, as the following
examples illustrate, it is not as well-behaved as one might initially expect.
Example 9.2. Suppose that fn : (0, 1) → R is defined by
n
fn (x) = .
nx + 1
Then, since x 6= 0,
1 1
lim fn (x) = lim = ,
n→∞ n→∞ x + 1/n x

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