Basel problem: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Barry R. Smith
m (Basel Problem moved to Basel problem: capitalization problem)
imported>Barry R. Smith
mNo edit summary
Line 1: Line 1:
{{subpages}}
{{subpages}}


The Basel problem is a question about [[infinite series]], first formulated by [[Pietro Mengoli]] in 1650<ref>See Andre Weil, ''Basic Number Theory'', p.184</ref>  Many of the most talented and influential mathematicians of that time attempted and failed to solve the problem.  Its solution was one of the early triumphs of [[Leonard Euler]], and immediately made him renown in European mathematical circles.   
The '''Basel problem''' is a question about [[infinite series]], first formulated by [[Pietro Mengoli]] in 1650<ref>See Andre Weil, ''Basic Number Theory'', p.184</ref>  Many of the most talented and influential mathematicians of that time attempted and failed to solve the problem.  Its solution was one of the early triumphs of [[Leonard Euler]], and immediately made him renown in European mathematical circles.   


Mengoli knew that the series of positive numbers <math>\scriptstyle \sum_{n=1}^{\infty} n^{-2} </math> is [[convergent]].  In the Basel problem, he asked for its sum.  Euler showed that
Mengoli knew that the series of positive numbers <math>\scriptstyle \sum_{n=1}^{\infty} n^{-2} </math> is [[convergent]].  In the Basel problem, he asked for its sum.  Euler showed that

Revision as of 20:17, 28 March 2008

This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

The Basel problem is a question about infinite series, first formulated by Pietro Mengoli in 1650[1] Many of the most talented and influential mathematicians of that time attempted and failed to solve the problem. Its solution was one of the early triumphs of Leonard Euler, and immediately made him renown in European mathematical circles.

Mengoli knew that the series of positive numbers is convergent. In the Basel problem, he asked for its sum. Euler showed that

He actually showed considerable more than this, evaluating the sums of several related series which, from a modern viewpoint, are values of the Riemann zeta function at positive even integers. His argument was highly non-rigorous, assuming that results known to be true for finite sums and products also extend to infinite series and products. Even so, he had already calculated its value to 20 decimal places[2], and up to this precision, the value matched his claimed . Thus, mathematicians of the time who disbelieved the validity of his argument could not argue that the value he had found was incorrect.

  1. See Andre Weil, Basic Number Theory, p.184
  2. See William Dunham, Journey Through Genius