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Basel Problem

By Moloy De posted Fri July 07, 2023 09:20 PM

  
The Basel Problem is the name given to the problem concerning finding the infinite sum of the reciprocal of squares of natural numbers. It was first proposed by Pietro Mengoli and finally solved by the one and only Leonhard Euler. So this problem is named after Euler’s hometown Basel.
Let's start with the Maclaurin series of sin which expresses sinx as a sum of polynomials.
Euler expressed sinx as a product of polynomials too.
Now equating the coefficients of x³ from both the infinite sum and the infinite product Euler gets,
Voila! We have thus arrived at the famous result obtained by Euler.

QUESTION I: Does the infinite sum of the reciprocals of natural numbers converge?

QUESTION II: Do the infinite sums of the reciprocals of other powers of natural numbers converge?

REFERENCE: Basel Problem Blog

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