Show that $\sum_{n=1}^{\infty}{\frac{1}{n^2}}=\frac{\pi^2}{6}$

Question

Show that $$\sum_{n=1}^{\infty}{\frac{1}{n^2}}=\frac{\pi^2}{6}$$ Anyone can help ?

Answer 1

This is known as the Basel problem and was first solved by Euler. His derivation (shown at above link) does some clever manipulations with the power series expansion of $\frac{\sin(x)}{x}$.

A more advanced proof uses Fourier transforms and Parseval's identity for the function $f(x)=x$.

The link also gives a rigorous but elementary proof, requiring only trigonometric identities and binomial coefficients, together with the "pinching lemma", but no calculus.