The Basel problem

Solved in 1734 by Leonhard Euler.

Pietro Mengoli posed the problem in 1650. It is named after Basel, the hometown of Euler and of the Bernoulli family, who attacked it without success. In 1734 Euler found the value ${\pi }^2/6$ by treating $\sin x/x$ as an infinite product over its zeros and comparing coefficients, a step that was not justified at the time but was later made rigorous through the Weierstrass factorization theorem. Euler himself gave an accepted proof in 1741. Many independent proofs are now known, using Fourier series, double integrals or complex analysis, and Euler went on to evaluate $\sum 1/n^{2k}$ for every positive integer $k$.

References

• Basel problem (Wikipedia)