Definite Integral of Uniformly Convergent Series of Continuous Functions

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Theorem

Let $\left\langle{f_n}\right\rangle$ be a sequence of real functions.

Let each of $\left\langle{f_n}\right\rangle$ be continuous on the interval $\left[{a \,.\,.\, b}\right)$.



Let the series:

$\displaystyle f \left({x}\right) := \sum_{n \mathop = 1}^\infty f_n \left({x}\right)$

be uniformly convergent for all $x \in \left[{a \,.\,.\, b}\right]$.


Then:

$\displaystyle \int_a^b f \left({x}\right) \ \mathrm dx := \sum_{n \mathop = 1}^\infty \int_a^b f_n \left({x}\right) \ \mathrm dx$


Proof

Define $S_N(x) = \sum_{n = 1}^N f_n(x)$. We have $$\left\vert \int_a^b f(x) dx - \sum_{n = 1}^N \int_a^b f_n(x) dx \right\vert = \left\vert \int_a^b f(x) - S_N(x) dx \right\vert \leq (b - a) \sup_{x \in [a, b]} \vert f(x) - S_N(x) \vert \to 0$$ as $N \rightarrow + \infty$.

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