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I'm searching for a proof that $\pi$ is irrational using a series representation for $\pi$, but can't find it.

However, on this wikipedia page show's that Apery's proof on the irrationality of $\zeta(3)$ can be simplified to apply on $\zeta(2)$ which is better than what i'm looking for because it shows that $\pi^2$ is irrational. But I can't find this proof either. If it's was done, maybe no one ever published.

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  • $\begingroup$ Check out van der Poorten's survey on Apery's proof "A proof that Euler missed" in Math. Intelligencer, as far as I remember it does $\zeta(2)$ as well. $\endgroup$
    – Piotr Achinger
    Commented May 16, 2019 at 3:33

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Beukers gave a proof of the irrationality of $\zeta(2)$ and $\zeta(3)$ using Apery's ideas, expressed in integral form here. This was published in 1979.

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  • $\begingroup$ I've seen that but I'm interessed in proof's that uses fast convergent series. $\endgroup$
    – Pinteco
    Commented May 16, 2019 at 3:24

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