sequences and series - How to prove this identity $pi=sumlimits_{k=-infty}^{infty}left(frac{sin(k)}{k}right)^{2};$?

How to prove this identity?

$$\pi=\sum_{k=-\infty}^{\infty}\left(\dfrac{\sin(k)}{k}\right)^{2}\;$$

I found the above interesting identity in the book $\bf \pi$ Unleashed.

Does anyone knows how to prove it?

Thanks.

Answer

Find a function whose Fourier coefficients are $\sin{k}/k$. Then evaluate the integral of the square of that function.

To wit, let

$$f(x) = \begin{cases} \pi & |x|<1\\0&|x|>1 \end{cases}$$

Then, if

$$f(x) = \sum_{k=-\infty}^{\infty} c_k e^{i k x}$$

then

$$c_k = \frac{1}{2 \pi} \int_{-\pi}^{\pi} dx \: f(x) e^{i k x} = \frac{\sin{k}}{k}$$

By Parseval's Theorem:

$$\sum_{k=-\infty}^{\infty} \frac{\sin^2{k}}{k^2} = \frac{1}{2 \pi} \int_{-\pi}^{\pi} dx \: |f(x)|^2 = \frac{1}{2 \pi} \int_{-1}^{1} dx \: \pi^2 = \pi$$

ADDENDUM

This result is easily generalizable to

$$\sum_{k=-\infty}^{\infty} \frac{\sin^2{a k}}{k^2} = \pi\, a$$

where $a \in[0,\pi)$, using the function

$$f(x) = \begin{cases} \pi & |x|a \end{cases}$$