Basel problem

The Basel problem is a mathematical problem that was unsolved for a long time and with the initially dealt primarily Basle mathematician. It is the question of the sum of the reciprocal squares, after the value of the series

Solution attempts

1644 asked the Italian Pietro Mengoli if this sum converges, and if so, to what value, this question but could not answer. A little later learned the Basel mathematician Jacob Bernoulli by this problem, but found no solution (1689 ). Then, several mathematicians tried on the question, but all were unsuccessful. 1726 began Leonhard Euler, Basel also a mathematician and student of Jakob Bernoulli's brother, John, to deal with the problem. 1735, he found the solution and published it in his work " De Summis Serierum Reciprocarum ".

Euler's solution

For his original solution Euler considered the Taylor series of cardinal sine function, ie

And she sat with the product representation of that function the same.

Multiplying out and sort one obtains:

And it concluded its Euler solution:

Also generalized Euler problem. He examined for the Riemann ζ - function later called function

And found a general closed expression for all even natural arguments, namely

Where the- th Bernoulli number represents. A general formula for odd natural arguments ( see, for example apery constant) is unknown.