# Evaluate $\int_{-\pi}^\pi \big|\sum^\infty_{n=1} \frac{1}{2^n} e^{inx}\big|^2 \operatorname d\!x$

I am trying to solve exercises for the coming exam, and I am stuck on this exercise:

Evaluate the integral $$\int_{-\pi}^\pi \Big|\sum^\infty_{n=1} \frac{1}{2^n} \mathrm{e}^{inx}\,\Big|^2 \operatorname d\!x$$

A page before it intoduced the Parseval's identity, so I guess it is related to it.

I tried to solve it, but whan ever it try is bad.

Can you please give me some hints? Thanks!

• I didn't know that double dollar $$_$$ works in the title too. It gets beautiful. – Billy Rubina Feb 1 '14 at 23:06
• You have to apply Parseval's identity to the function $g$ defined by : $$\displaystyle \forall x \in \mathbb{R}, \; g(x) = \sum_{n=1}^{+\infty} \frac{1}{2^{n}} e^{inx}$$ which means that you (almost) only have to compute the Fourier coefficients of $g$, which shouldn't be too hard, I think. – jibounet Feb 1 '14 at 23:09
• Duplicate – Mhenni Benghorbal Feb 2 '14 at 2:23
• @PristineKavalostka This is explicitly discouraged, though. – AlexR Feb 24 '14 at 23:26

Define $$g(x) = \sum\limits_{n = 1}^{\infty} \frac{e^{inx}}{2^n}$$

This is absolutely convergent everywhere, and defines a continuous function (by, for example, the Weierstrass $M$-test). The Fourier coefficients of $g$ are easy to compute: Define $$c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} g(x) e^{-inx} dx$$

By orthogonality of the functions $\{e^{inx}\}$, we find that

$$c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} \sum_{k = 1}^{\infty} \frac{e^{i(k - n)x}}{2^k} dx = \frac{1}{2\pi} \frac{2\pi}{2^n} = \frac{1}{2^n}$$ Then Parseval's identity shows that

$$\int_{-\pi}^{\pi} |g(x)|^2 dx = 2\pi \sum\limits_{n = 1}^{\infty} |c_n|^2 = 2\pi \sum_{n = 1}^{\infty} 4^{-n} = \frac{2\pi}{3}$$

• Is there something you don't know? You are a genius!!! Thank you!!! – Billie Feb 1 '14 at 23:26
• Can you please explain me how did you get: $c_n = \frac{1}{2\pi} \int_{- \\pi}^{\pi} \sum_{k = 1}^{\infty} \frac{e^{i(k - n)x}}{2^k} dx = \frac{1}{2\pi} \frac{2\pi}{2^k} = \frac{1}{2^k}$ – Billie Feb 1 '14 at 23:32
• It should be $\frac{1}{2\pi}\frac{2\pi}{2^n} = \frac{1}{2^n}$. – TonyK Feb 1 '14 at 23:34
• @TonyK Still don't get how he got to it – Billie Feb 1 '14 at 23:34
• $\int_{-\pi}^{\pi} e^{i m x} dx = 2\pi$ if $m = 0$, and $0$ otherwise. – user61527 Feb 1 '14 at 23:34

\begin{align} \int_{-\pi}^\pi \Big|\sum^\infty_{n=1} \frac{1}{2^n} \mathrm{e}^{inx}\Big|^{\,2} \operatorname d\!x &= \int_{-\pi}^\pi \sum^\infty_{n=1} \frac{1}{2^n} \mathrm{e}^{inx} \overline{\sum^\infty_{m=1} \frac{1}{2^m} \mathrm{e}^{imx}} \operatorname d\!x= \sum_{m,n=1}^\infty \int_{-\pi}^\pi\frac{1}{2^{m+n}} \mathrm{e}^{i(m-n)x}\,dx \\ &=\sum_{n=1}^\infty\frac{2\pi}{2^{2n}}=\frac{\frac{2\pi}{4}}{1-\frac{1}{4}}=\frac{2\pi}{3}, \end{align} since $$\int_{-\pi}^\pi\frac{1}{2^{m+n}} \mathrm{e}^{i(m-n)x}\,dx=\frac{2\pi}{2^{m+n}}\,\delta_{m,n}.$$

• your second equation misses the constant $\frac{2\pi}{2^{m+n}}\delta_{m,n}$ ($\int e^{i(m-n)x} dx = 2\pi\delta_{m,n}$). – AlexR Feb 24 '14 at 23:27


Hint:

$$\sum^\infty_{n=1} 2^{-n} e^{inx} = \frac{e^{ix}/2}{1-e^{ix}/2}$$

After squaring and simplifying, contour integration around $|z|=1$ should do the trick.