Sunday, 6 September 2015

sequences and series - Find the sum of sum1/(k2a2) when $0



So I have been trying for a few days to figure out the sum of



S=k=11k2a2 where a(0,1). So far from my nummerical
analysis and CAS that this sum equals



S=12a[1aπcot(aπ)]



But I have not been able to prove this yet. Anyone know how? My guess is that the

sum of this series is related to fourier-series but nothing particalr comes to mind.



For the easy values, I have been able to use telescopic series, and a bit of algebraic magic, but for the general case I am stumpled. Anyone have any ideas or hints? Cheers.


Answer



This question was settled in the Mathematics chatroom, but I'll put up the solution here for reference.



Starting with the infinite product



sinπxπx=k=1(1x2k2)




taking the logarithm of both sides gives



log(sinπxπx)=log(k=1(1x2k2))=k=1log(1x2k2)



Differentiation gives



πxsinπx(cosπxxsinπxπx2)=k=12xk2(1x2k2)



which simplifies to




πcotπx1x=2xk=11k2x2



or



k=11k2x2=12x2πcotπx2x


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