I was studying electromagnetism and followed 'Introduction to Electromagnetism' by David Griffiths. During his derivation of the solution to Laplace's equation in ch. 3.3, he derives the equation $$V(x,y)=\frac{4 V_0}{\pi} \sum_{n=1,3,5...} \left[\frac{1}{n}\text{e}^{-\frac{n \pi x}{a}} \text{sin}\left(\frac{n \pi y}{a}\right) \right].$$
It is stated that the infinite series can be summed explicitly and the complete expression evaluates to $V(x,y)=\frac{2 V_0}{\pi} \text{tan}^{-1}\left(\frac{\text{sin}(\frac{\pi y}{a})}{\text{sinh}(\frac{\pi x}{a})}\right)$, and I was wondering how he derives this results. We obviously have to determine the convergence of the series, and I assume we can use Parseval's theorem, but some help would be greatly appreciated.
Sincerely Rasmus