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: