Two remarkable sums due to Ramanujan-Part I

In this article we prove two remarkable results due to Ramanujan, namely

$\begin{align*} \sum_{n=1}^\infty \frac{n }{e^{2 \pi n}-1}&=\frac{1}{24}-\frac{1}{8 \pi } \end{align*}$

and

$\begin{align*} \sum_{n=1}^\infty \frac{1}{\sinh^2 \pi n}&= \frac{1}{6}-\frac{1}{2 \pi} \end{align*}$

To this end we rely on the functional equation that the Dedekind´s eta

function obeys, previously proved (here):

$\begin{align*} \eta\left( -\frac{1}{\tau}\right)&=\sqrt{-i \tau}\,\,\eta\left( \tau\right) \end{align*}$

Click here for the proof.

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