Let $\displaystyle \zeta(s)=\sum_{n>1}\frac{1}{n^s}$ denote the Riemann Zeta function for complex $s$ such that $\Re(s)>1$. While proving that $\zeta(3)$ is irrational, Roger Apéry found$^1$ the remarkable accelerated series
$$ \zeta(3)=\frac{5}{2} \sum_{k=1}^{\infty}{\frac{ (-1)^{k-1}} {\binom {2k}{k}k^{3}}}. $$
His proof of this identity is outlined in Van der Poorten's "A Proof that Euler Missed...", and a breakdown of this can be found in this Math.SE question. Another, different proof of this identity using creative telescoping can be found in Jack d'Aurizio's textbook.
Inspired by another Math.SE post regrouping different proofs for $\displaystyle \zeta(2)=\frac{\pi^2}{6}$, I became interested in collecting various proofs for this identity above in hopes of adapting this series to higher zeta constants. What other proofs are there for this famous identity?
$^1$ In fact, it was the Norwegian mathematician Hjortnaes who first found this identity. His paper however has been lost since and the original proof is unfortunately impossible to find on the Internet.