# Integrating squared absolute value of a complex sequence

I was reading through my book in complex analysis and i encountered this problem.
Given, $F=\sum_{n=0}^{\infty} a_nX^n$ is a convergent power series with radius of convergence R.
We are asked to show that for every 0$\leq$r$<$R that $$\int_0^{2\pi}\mid F(re^{it})\mid^2\mathrm{d}t={2\pi}\sum_{n=0}^{\infty}\mid a_n\mid^2r^{2n}$$
This is not my homework but after some effort I have no clue how to solve this one. The assignment was in the chapter about convergent power series and there is nothing about integrating, thus I assume it should be solvable by real analysis calculus. I would appreciate if someone would help me with this one.
Regards, Raxel.

#### Solutions Collecting From Web of "Integrating squared absolute value of a complex sequence"

$$|F(re^{it})|^2=\sum_{m=0}^\infty a_mr^me^{imt}\times\sum_{n=0}^\infty \bar{a}_nr^ne^{-int}=\sum_{m,n=0}^\infty a_m\bar{a}_nr^{m+n}e^{i(m-n)t}.$$

Integrating both sides over $[0,2\pi]$ and interchanging the order of integration and summation, and noting that the value of $\int_0^{2\pi}e^{i(m-n)t}dt$ is $2\pi$ when $m=n$ and $0$ when $m\ne n$, then you will obtain the wanted answer. Please check that each step is legal by using the fact $r<R$.