How to prove this identity? π=∞∑k=−∞(sin(k)k)2

I found the above interesting identity in the bookπ Unleashed.Does anyone knows how to prove it?

Thanks.

**Answer**

Find a function whose Fourier coefficients are sink/k. Then evaluate the integral of the square of that function.

To wit, let

f(x)={π|x|<10|x|>1

Then, if

f(x)=∞∑k=−∞ckeikx

then

ck=12π∫π−πdxf(x)eikx=sinkk

∞∑k=−∞sin2kk2=12π∫π−πdx|f(x)|2=12π∫1−1dxπ2=π

**ADDENDUM**

This result is easily generalizable to

∞∑k=−∞sin2akk2=πa

where a∈[0,π), using the function

f(x)={π|x|<a0|x|>a

**Attribution***Source : Link , Question Author : Neves , Answer Author : Ron Gordon*