Basel problem

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


Though Mengoli verified the Wallis formulae for π, it did not occur to him that π 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


The value, 1.6449340668482264365… could then be computed to almost as many decimal places as were known of π. See value of the Riemann zeta functionDlmfDlmfMathworldPlanetmath at s=2 (


  • 1 Ed Sandifer, “Euler’s Solution of the Basel Problem - The Longer Story”. Danbury, Connecticut: Western Connecticut State University (2003)
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