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- #1

- Jan 17, 2013

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This thread will be dedicated for a trial to prove the following

In this paper the authors give solutions to the sum and others , but the process is quite complicated and uses contour integration , Fourier series and Parseal's thoerem , .... .

I believe we can solve it using elementary methods .

\(\displaystyle \sum_{k\geq 1} \frac{H^2_k}{k^2}=\frac{17}{4}\zeta(4)=\frac{17\pi^4}{360}\)

\(\displaystyle \mbox{where }\,\,H^2_k =\left( 1+\frac{1}{2}+\frac{1}{3}+\cdots \frac{1}{k}\right)^2\)

\(\displaystyle \mbox{where }\,\,H^2_k =\left( 1+\frac{1}{2}+\frac{1}{3}+\cdots \frac{1}{k}\right)^2\)

In this paper the authors give solutions to the sum and others , but the process is quite complicated and uses contour integration , Fourier series and Parseal's thoerem , .... .

I believe we can solve it using elementary methods .

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