- #1
- 19,513
- 25,502
Prove
$$
\zeta(2) = \sum_{n\in \mathbb{N}}\dfrac{1}{n^2} = \dfrac{\pi^2}{6}
$$
by evaluating
$$
\int_0^1\int_0^1\dfrac{1}{1-xy}\,dx\,dy
$$
twice: via the geometric series and via the substitutions ##u=\dfrac{y+x}{2}\, , \,v=\dfrac{y-x}{2}##.
$$
\zeta(2) = \sum_{n\in \mathbb{N}}\dfrac{1}{n^2} = \dfrac{\pi^2}{6}
$$
by evaluating
$$
\int_0^1\int_0^1\dfrac{1}{1-xy}\,dx\,dy
$$
twice: via the geometric series and via the substitutions ##u=\dfrac{y+x}{2}\, , \,v=\dfrac{y-x}{2}##.