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It is known that there are many formulas or sequences that give the exact value of $\pi$,but is there any proof that unlimited of them exist? Conditioned that when u plot it in a graph the function should not be perfectly equal

Nimish
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    What is your definition of "equal" for two formulae? – Michael Burr Jun 04 '17 at 10:54
  • You can rewrite the same formule an in infinite amount of ways, so yes. –  Jun 04 '17 at 10:54
  • Of course without rewriting ,there must be some kind of proof that infinite expressions exist – Nimish Jun 04 '17 at 11:19
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    For that you have to define what you mean by rewriting. Maybe someone considers $\zeta(2)$ to be just a rewriting of $\pi^2/6$ and similar; then infinitylord's answer doesn't give us infinitely many different expressions. – Wojowu Jun 04 '17 at 11:32
  • by not rewriting I mean not rearranging the terms of an expression – Nimish Jun 04 '17 at 11:36
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    If I have a formula $\phi$ for $\pi$, then also $\phi+0$ will give you $\pi$, and also $\phi+0+0$. So there are infinitely many. Without an explanation for what formulas you will accept as different there is not much that can be said. Not rewriting is not very helpful. Any proof for that $\phi$ will give you $\pi$ is essentially a proof that $\phi$ can be rewritten as the formula that initially defined $\pi$. – M. Winter Jun 04 '17 at 11:41

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The values for $\zeta(2k)$ are known, and are of the form $$\zeta(2k) = \alpha_{2k} \pi^{2k}$$ So $$(\frac{\zeta(2k)}{\alpha_{2k}})^{\frac1{2k}} = \pi $$

infinitylord
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There are infinitely many sequences converging to $\pi$. For example, take any sequence $x_n\to\pi$. Then the sequence

$$x_n+\frac cn$$

will converge to $\pi$ too for any $c\in\Bbb R$.

M. Winter
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