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 intervalI .
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

?