Easy question about finite energy due to convergence The infinite-length sequence $x_1[n]$ defined by 
\begin{multline}
x_1[n]=
\begin{cases}
\dfrac{1}{n}& \text{if $n \geq $1},\
0& \text{if $n \leq $0}.
\end{cases}
\end{multline}
has an energy equal to 
$\mathcal{E _x {_1}} = \sum^\infty_{n=1}(\dfrac{1}{n})^{2}$
which converges to $\pi^2/6$ indicating that $x_1[n]$ has finite energy.
I don't get where we find $\pi^2/6$. It would be great if anyone can help me out.

The sum of the series $\displaystyle\sum^\infty_{n=1}(\dfrac{1}{n})^{2}=\dfrac{\pi^2}{6}$ is a classical result due to Euler. Several proofs are given in the answers to Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem).

PS. Here Robin Chapman collects 14 proofs.

PPS. The improper double integral $$\int_{0}^{1}\int_{0}^{1}\left(\dfrac{1}{1-xy}\right) \mathrm{d}x\mathrm{d}y=\int_{0}^{1}\int_{0}^{1}\left(\sum_{n=1}^{\infty }\left( xy\right)^{n-1}\right) \mathrm{d}x\mathrm{d}y=\sum^\infty_{n=1}\dfrac{1}{n^2} =\dfrac{\pi^2}{6}=\zeta(2)$$ is finite, as pointed out in Proofs from THE BOOK by Martin Aigner and Günter Ziegler. The original article by Tom Apostol is here.
