Definite Integral of Uniformly Convergent Series of Continuous Functions

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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 $x \in \closedint a b$.


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


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 $N \to +\infty$