# Looking for a proof that $\pi$ is irrational using a series representation.

I'm searching for a proof that $$\pi$$ is irrational using a series representation for $$\pi$$. I've seen that Apery proved that $$\zeta(2)$$ is irrational by using the series \begin{align} \zeta(2) = \frac{\pi^2}{6} = 3 \sum_{n=1}^{\infty} \frac{1}{n^2\binom{2n}{n}} \end{align} but can't find the proof. Since proving that $$\pi^2$$ is irrational also shows that $$\pi$$ is irrational this proof is interesting.

This is the only proof I've heard of (but can't find it) that uses a series for $$\pi$$.

• What about these? – Rhys Hughes Dec 31 '18 at 19:29
• But is there any proof that $\pi$ is irrational using those series? – Pinteco Dec 31 '18 at 19:55