Apéry's Theorem

Theorem
Apéry's constant:
 * $\map \zeta 3 = \ds \sum_{n \mathop = 1}^\infty \frac 1 {n^3}$

is irrational.

Proof
We have:


 * $\dfrac 6 {\map \zeta 3} = 5 - \cfrac {1^6} {117 - \cfrac {2^6} {535 - \cfrac {\dotsb} {\dotsb - \cfrac {n^6} {34 n^3 + 51 n^2 + 27 n + 5} - \dotsb } } }$

Also known as
Some sources present the name of this without the accent: Apery's theorem.