From charlesreid1


MediaWiki Math

To make a multiline equation in MediaWiki, use the following syntax:

\pi^{-1} &=& \dfrac{\sqrt{8}}{99^2} \sum_{k \geq 0} \dfrac{ (4k)! }{ (4^k k!)^4 } \dfrac{1103 + 26390 k}{ 99^4k } \\
\pi^{-1} &=& \dfrac{1}{\pi}

\pi^{-1} &=& \dfrac{\sqrt{8}}{99^2} \displaystyle{ \sum_{k \geq 0} \dfrac{ (4k)! }{ (4^k k!)^4 } \dfrac{1103 + 26390 k}{ 99^4k } } \\
\pi^{-1} &=& \dfrac{1}{\pi}

Floating Point Numbers

What every computer scientist should know about floating point numbers:


Infinite Series of 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}


\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