Proof of Uniform Convergence on I for $\sum_{n=1}^{\infty}f_n(x)$

[burak100]

Homework Statement

$I = [0, \frac{\Pi}{2}]$ is an interval, and $\lbrace f_n(x)\rbrace_{n=1}^{\infty}$ is sequence of continuous function. $\sum_{n=1}^{\infty}f_n(x)$ converges uniformly on the interval$I$ .
Show that holds

$\int_{0}^{\frac{\Pi}{2}} \sum_{n=1}^{\infty} f_n(x)dx = \sum_{n=1}^{\infty} \int_{0}^{\frac{\Pi}{2}} f_n(x)dx$

Homework Equations

The Attempt at a Solution

I`m not familiar that how we show this with using uniformly converge?
pls help, I will be appreciate...

 

[burak100]

can we say that?

If $\sum_{n=1}^{\infty}f_n(x)$ converges uniformly on $I=[0, \frac{\Pi}{2}]$,
then we can write,

$\int_0^{\frac{\Pi}{2}} \sum_{n=1}^{\infty}f_n(x)dx = \int_0^{\frac{\Pi}{2}} \lim_{\frac{\Pi}{2} \rightarrow \infty} S_n(x)dx = \lim_{\frac{\Pi}{2} \rightarrow \infty} \int_0^{\frac{\Pi}{2}} S_n(x)dx$
$~~~=\sum_{n=1}^{\infty} \int_0^{\frac{\Pi}{2}} f_n(x)dx$

?