# 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)

## References

- 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 |