Page 185 - DMTH401_REAL ANALYSIS

Richa Nandra, Lovely Professional University Unit 14: Sequences and Series of Functions

Unit 14: Sequences and Series of Functions Notes

CONTENTS
Objectives
Introduction
14.1 Sequences of Functions
14.2 Uniform Convergence

14.3 Series of Functions
14.4 Summary
14.5 Keywords
14.6 Review Questions
14.7 Further Readings

Objectives

After studying this unit, you will be able to:

 Define sequence and series of functions
 Distinguish between the pointwise and uniform convergence of sequences and series of
functions
 Know the relationship of uniform convergence with the notions of continuity,
differentiability, and integrability

Introduction

In earlier unit, we have studied about, convergence of the infinite series of real numbers. In this
unit, we want to discuss sequences and series whose members are functions defined on a subset
of the set of real numbers. Such sequences or series are known as sequences or series of real
functions. You will be introduced to the concepts of pointwise and uniform convergence of
sequences and series of functions. Whenever they are convergent, their limit is a function called
limit function. The question arises whether the properties of continuity, differentiability,
integrability of the members of a sequence or series of functions are preserved by the limit
function. We shall discuss this question also in this unit and show that these properties are
preserved by the Uniform convergence and not by the pointwise convergence.

14.1 Sequences of Functions

As you have studied that a sequence is a function from the set N of natural numbers to a set B. In
that unit, sequences of real numbers have been considered in detail. You may recall that for
sequences of real numbers, the set B is a sub-set of real numbers. If the set B is the set of real
functions defined on a sub-set A of R, we get a sequence called sequence of functions. We define
it in the following way:
Definition 1: Sequence of Functions

Let A be a non-empty sub-set of R and let B be the set of all real functions each defined on A.
A mapping from the set N of natural numbers to the set B of real functions is called a sequence
of functions.

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