How to prove this identity π=∞∑k=−∞(sin(k)k)2\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;?

How to prove this identity? π=k=(sin(k)k)2
I found the above interesting identity in the book π 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=12πππdxf(x)eikx=sinkk

By Parseval’s Theorem:

k=sin2kk2=12πππdx|f(x)|2=12π11dxπ2=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

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