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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 $\pi$.

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

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