# Converges uniformly

1. Sep 10, 2011

### burak100

1. The problem statement, all variables and given/known data
$I = [0, \frac{\Pi}{2}]$ 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

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

2. Relevant equations

3. The attempt at a solution
I`m not familiar that how we show this with using uniformly converge????
pls help, I will be appreciate...

2. Sep 10, 2011

### 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,

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

??????