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