Tuesday, 25 December 2018

sequences and series - How to prove this identity pi=sumlimitsinftyk=inftyleft(fracsin(k)kright)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



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


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