Parseval's Identity (Integral)
Solution 1:
Here is how,
$$ \int_{-\pi}^{\pi}\left|\sum_{n=1}^{\infty}\frac{1}{2^{n}}e^{inx}\right|^{2}dx = \sum_{n=1}^{\infty}\frac{1}{2^{n}}\sum_{m=1}^{\infty}\frac{1}{2^{m}} \int_{-\pi}^{\pi} e^{i(n-m)x}dx .$$
Now, see here for details and how to finish the problem. Another related technique.