# Definite Integral of Uniformly Convergent Series of Continuous Functions

## Theorem

Let $\sequence {f_n}$ be a sequence of real functions.

Let each of $\sequence {f_n}$ be continuous on the interval $\hointr a b$.

Let the series:

$\ds \map f x := \sum_{n \mathop = 1}^\infty \map {f_n} x$

be uniformly convergent for all $x \in \closedint a b$.

Then:

$\ds \int_a^b \map f x \rd x = \sum_{n \mathop = 1}^\infty \int_a^b \map {f_n} x \rd x$

## Proof

Define $\map {S_N} x = \ds \sum_{n \mathop = 1}^N \map {f_n} x$.

We have:

 $\ds \size {\int_a^b \map f x \rd x - \sum_{n \mathop = 1}^N \int_a^b \map {f_n} x \rd x}$ $=$ $\ds \size {\int_a^b \paren {\map f x - \map {S_N} x} \rd x}$ $\ds$ $\le$ $\ds \paren {b - a} \sup_{x \mathop \in \closedint a b} \size {\map f x - \map {S_N} x}$ $\ds$ $\to$ $\ds 0$ as $N \to +\infty$

$\blacksquare$