Math

Euler

Infinite Series of Surprises: https://plus.maths.org/content/infinite-series-surprises

Basel Problem:

$ \sum_{k=1}^{\infty} \dfrac{1}{k^2} = \dfrac{\pi^2}{6} $

This proof extends to other even powers as well:

$ \sum_{k=1}^{\infty} \dfrac{1}{k^4} = \dfrac{\pi^4}{90} $

and

$ \sum_{k=1}^{\infty} \dfrac{1}{k^6} = \dfrac{\pi^6}{945} $

Then, in 1744, obtained:

$ \sum_{k=1}^{\infty} \dfrac{1}{k^26} = \dfrac{2^{24} 76977927 \pi^{26} }{27!} $

by the same method.

This principle solves

$ \sum_{k=1}^{\infty} = \dfrac{1}{k^{2n}} $

for natural numbers $ n $.

The corresponding set of problems for odd powers,

$ \sum_{k=1}^{\infty} \dfrac{1}{k^3} $

is still an open problem. The best Euler could do was:

$ \sum_{k=0}^{\infty} \dfrac{ (-1)^k }{ (2k+1)^3 } = 1 - \frac{1}{27} + \frac{1}{125} - \dots = \dfrac{ \pi^3 }{32} $

Translation of Euler's paper: Remarques sur un beau rapport entre les series des puissances tant directes que reciproques

http://eulerarchive.maa.org//docs/translations/E352.pdf