Basel problem

The Basel problem, first posed by Pietro Mengoli in 1644, asks for a finite formula for the infinite sum

$$\sum _{i=1}^{\mathrm{\infty }}\frac{1}{i^2}.$$

Though Mengoli verified the Wallis formulae for $\pi$, it did not occur to him that $\pi$ was also involved in the solution of this problem. Jakob Bernoulli also tried in vain to solve this problem. Even an approximate decimal value eluded contemporary mathematicians: an answer accurate to just five decimal places requires iterating up to at least $i=112000$, which without the aid of a computer was wholly impractical in Mengoli’s day. The problem was finally solved in 1741, when, after almost a decade of work, Leonhard Euler conclusively proved that

$$\frac{{\pi }^2}{6}=\zeta (2)=\sum _{i=1}^{\mathrm{\infty }}\frac{1}{i^2}.$$

The value, 1.6449340668482264365… could then be computed to almost as many decimal places as were known of $\pi$. See value of the Riemann zeta function at $s=2$ (http://planetmath.org/ValueOfTheRiemannZetaFunctionAtS2)

Title	Basel problem
Canonical name	BaselProblem
Date of creation	2013-03-22 18:05:22
Last modified on	2013-03-22 18:05:22
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	5
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A25