Summing some Eisenstein series

In this blog entry we will prove the following three beautifull infinte series


\begin{align*} &\sum_{n=1}^{\infty} \frac{n^{5} }{ e^{2 \pi n }-1 }=\frac{1}{504}\\ &\sum_{n=1}^{\infty} \frac{n^{9} }{ e^{2 \pi n }-1 }=\frac{1}{264}\\ &\sum_{n=1}^{\infty} \frac{n^{13} }{ e^{2 \pi n }-1 }=\frac{1}{24}\\ \end{align*}


Click here for the proof.

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