## Basel problem |

Part of |

mathematical constant |
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3.1415926535897932384626433... |

Uses |

Properties |

Value |

People |

History |

In culture |

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The **Basel problem** is a problem in ^{[1]} and read on 5 December 1735 in *The Saint Petersburg Academy of Sciences*^{[2]} Since the problem had withstood the attacks of the leading

The Basel problem asks for the precise

- .

The sum of the series is approximately equal to **1.644934**.^{[3]} The Basel problem asks for the *exact* sum of this series (in ^{2}/6 and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, although he was later proven correct, and it was not until 1741 that he was able to produce a truly rigorous proof.

- euler's approach
- the riemann zeta function
- a rigorous proof using fourier series
- another rigorous proof using parseval's equality
- cauchy's proof
- other identities
- see also
- references
- notes
- external links

Euler's original derivation of the value π^{2}/6 essentially extended observations about finite

Of course, Euler's original reasoning requires justification (100 years later,

To follow Euler's argument, recall the

Dividing through by x, we have

Using the ^{[4]}

If we formally multiply out this product and collect all the *x*^{2} terms (we are allowed to do so because of *x*^{2} coefficient of sin *x*/*x* is ^{[5]}

But from the original infinite series expansion of sin *x*/*x*, the coefficient of *x*^{2} is −1/3! = −1/6. These two coefficients must be equal; thus,

Multiplying both sides of this equation by −π^{2} gives the sum of the reciprocals of the positive square integers.

This method of calculating is detailed in expository fashion most notably in Havil's *Gamma* book which details many ^{[6]}

Using formulas obtained from ^{[7]} this same approach can be used to enumerate formulas for the even-indexed

For example, let the partial product for expanded as above be defined by . Then using known

and so on for subsequent coefficients of . There are

which in our situation equates to the limiting recurrence relation (or

Then by differentiation and rearrangement of the terms in the previous equation, we obtain that

By Euler's proof for explained above and the extension of his method by elementary symmetric polynomials in the previous subsection, we can conclude that is *always* a

Other Languages

العربية: معضلة بازل

azərbaycanca: Bazel problemi

català: Problema de Basilea

Deutsch: Basler Problem

español: Problema de Basilea

français: Problème de Bâle

한국어: 바젤 문제

हिन्दी: बेसल समस्या

italiano: Problema di Basilea

עברית: בעיית בזל

magyar: Basel-probléma

Nederlands: Bazel-probleem

日本語: バーゼル問題

norsk: Baselproblemet

ਪੰਜਾਬੀ: ਬੇਸਲ ਸਮੱਸਿਆ

polski: Problem bazylejski

português: Problema de Basileia

русский: Ряд обратных квадратов

slovenščina: Baselski problem

српски / srpski: Базелски проблем

svenska: Baselproblemet

ไทย: ปัญหาบาเซิล

Türkçe: Basel problemi

українська: Ряд обернених квадратів

اردو: مسئلہ بازیل

中文: 巴塞尔问题