Apéry's constant

Apéry's constant is the value $\zeta(3)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k^3},$ where $\zeta$ denotes the Riemann zeta function. This constant is notable because it is known in general that for integers $n$, $\zeta(2n)$ is a rational multiple of $\pi$ but no general formula exists for values $\zeta(2n+1)$. Hence it became a notoriously open problem to find $\zeta$ at odd integers. One of the first results in this area is the following theorem by Apéry.

Theorem: The number $\zeta(3)$ is irrational.

Proof: █