# Convergence of Functions

Chapter

## Abstract

In many situations we have a sequence of functions f n that converges to some function f and f is not easy to study directly. Can we use the functions f n to get some information about f? For instance, if the f n are continuous, is f necessarily continuous? Another question that often comes up is: can I compute $$\int_{a}^{b} f\,dx$$ using $$\int_{a}^{b} f_{n}\,dx$$? More precisely, is it true that
$$\lim_{n\to\infty}\int_a^b f_n\,dx=\int_a^b f\,dx?$$
We can rewrite the question as
$$\lim_{n\to\infty}\int_a^b f_n\,dx=\int_a^b \lim_{n\to\infty}f_n\,dx?$$
In other words, can we interchange the limit and the integral? We will give some partial answers to these questions in this chapter. First, we need to define what we mean by the convergence of a sequence of functions. There are many different ways a sequence of functions can converge. In this chapter we will just consider two of them.

## Keywords

Power Series Triangle Inequality Uniform Convergence Fundamental Theorem Function Sine