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A beautiful sum!

\[\sum_{n=1}^{\infty} \left(\frac{\displaystyle \psi(n)-\psi\left(n-\frac{1}{2}\right)}{\displaystyle \binom{4n-2}{2n-1}}+\frac{\displaystyle \psi\left(n+\frac{1}{2}\right)-\psi(n)}{\displaystyle \binom{4n}{2n}}\right) = \frac{\pi}{3\sqrt{3}}+\frac{1}{75}\left(\log(1073741824)-3\sqrt{5} \pi^2+\pi\left(\Re \{{4 i \sqrt{5} \log(5- \sqrt{5} + i (\sqrt{15}- \sqrt{3}) ) - 2 i \sqrt{5} \log(5 +\sqrt{5}+ i (\sqrt{3} + \sqrt{15}))\}}+6\sqrt{5}\arctan\left(\sqrt{\frac{5}{3}}\right)-5\sqrt{3}\right)+\sqrt{5}\left(\log(79228162514264337593543950336) \log\left(\frac{3-\sqrt{5}}{2}\right) + \log(324518553658426726783156020576256)\log\left(\frac{3+\sqrt{5}}{2}\right)+24\Re \left\{\operatorname{Li}_2\left(\frac{\sqrt{3}+i}{\sqrt{3}+i\sqrt{5}}\right)-\operatorname{Li}_2\left(\frac{\sqrt{3}+i}{\sqrt{3}-i\sqrt{5}}\right)\right\}\right)\right)\]

Prove the above sum.

I found this sum unanswered in another mathematics forum. Can this be done without computational aid? Any ideas of solving this would be accepted.

Note by Aditya Kumar
1 year, 4 months ago

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It is known that \(\psi(n) = H_{n-1} - \gamma\) for all \(n \in \mathbb{N}\). With a little work, we can also show that \(\psi\left(n+\frac{1}{2}\right) = 2H_{2n} - H_n - \gamma - \ln(4)\).

Therefore, we can rewrite the sum as: \[\sum_{n=1}^{\infty} \left( \dfrac{2H_{n-1} - 2H_{2n-2} + \gamma + \ln(4) }{\binom{4n-2}{2n-1}} + \dfrac{2H_{2n} - 2H_n + \frac{1}{n} - \gamma - \ln(4)}{\binom{4n}{2n}} \right)\]

We can break these down into a few different sums, all of which obviously converge:

\[= \sum_{n=1}^{\infty} \dfrac{\gamma + \ln(4)}{\binom{4n-2}{2n-1}} - \sum_{n=1}^{\infty} \dfrac{\gamma + \ln(4)}{\binom{4n}{2n}} - \sum_{n=1}^{\infty} \dfrac{2H_{n-1} }{\binom{4n-2}{2n-1} } + \sum_{n=1}^{\infty} \dfrac{2H_{2n-2} }{ \binom{4n-2}{2n-1}} + \sum_{n=1}^{\infty} \dfrac{2H_{2n} }{\binom{4n}{2n} } - \sum_{n=1}^{\infty} \dfrac{2H_n}{\binom{4n}{2n} } + \sum_{n=1}^{\infty} \dfrac{1}{n \binom{4n}{2n}} \]

We can calculate the first two sums using the following McLauren Series:

\[f(x) = \sum_{k=1}^{\infty} \dfrac{(-1)^{k+1} x^{2k}}{\binom{2k}{k}} = \dfrac{4x \text{arcsinh}\left(\dfrac{x}{2}\right)}{(4+x^2)^{3/2}} + \dfrac{x^2}{4+x^2}\]

Then, \[\sum_{n=1}^{\infty} \dfrac{\gamma + \ln(4)}{\binom{4n-2}{2n-1}} - \sum_{n=1}^{\infty} \dfrac{\gamma + \ln(4)}{\binom{4n}{2n}} = (\gamma + \ln(4))f(1)\]

Of course that's just the beginning of the calculations, and it's already pretty complicated. I'm not sure how to calculate the sum of a harmonic number over a binomial coefficient.

Ariel Gershon - 1 year, 2 months ago

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