Math

Euler

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