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 $\hointr a b$.

This article, or a section of it, needs explaining. In particular: Investigation needed as to whether there is a mistake in 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) -- should it actually be a closed interval? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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 $N \to +\infty$

$\blacksquare$

Sources

• 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.3.1$: Properties of uniformly convergent series: Theorem $1.9 \ \text{(b)}$