Infinitely many expresions for $\pi$

Question

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

Answer 1

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

Answer 2

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 $c\in\Bbb R$.