Evaluate the sum
![\[\mathcal{S} = \sum_{n=1}^{\infty} \left( n \arccot n -1 \right)\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-3df1138bd9294b2d3757d3236db9cb20_l3.png "Rendered by QuickLaTeX.com")
Solution
Well, first of all we note that
We also recall the Fourier series expansion of 
which is no other than
(1) 
Hence
![\begin{align*} \mathcal{S} &= -\sum_{n=1}^{\infty} n \int_{0}^{1/n} \frac{x^2}{1+x^2} \, {\rm d}x\\ &=-\sum_{n=1}^{\infty} \int_{0}^{1} \frac{x^2}{x^2+n^2} \, {\rm d}x \\ &= \frac{1}{2} \int_{0}^{1} \left ( 1 - \pi x \coth \pi x \right ) \, {\rm d}x\\ &= \frac{1}{2} \int_{0}^{1} \left [ 1- \pi x \left ( 1 + \frac{2e^{-2\pi x}}{1-e^{-2\pi x}} \right ) \right ] \, {\rm d}x \\ &= \frac{1}{2} + \frac{17 \pi}{24} - \frac{1}{2} \log \left ( e^{2\pi} - 1 \right ) + \frac{1}{4\pi} {\rm Li}_2 \left ( e^{-2\pi} \right ) \end{align*}](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-b88a817104636666435178b5ea6a4b9d_l3.png "Rendered by QuickLaTeX.com")