Sum of $\pi_s$

$\displaystyle S = \sum_{n=1}^{\infty} \dfrac{1}{\pi^n} = \dfrac{1}{\pi} + \dfrac{1}{\pi^2} + \dfrac{1}{\pi^3} + \dfrac{1}{\pi^4} + \ldots \\ \displaystyle S \times \pi = 1 + \dfrac{1}{\pi} + \dfrac{1}{\pi^2} + \dfrac{1}{\pi^3} + \dfrac{1}{\pi^4} + \dfrac{1}{\pi^5} + \ldots \\ \displaystyle S\pi = 1 + S \\ \displaystyle S\pi - S = 1 \\ \displaystyle S(\pi - 1) = 1 \\ \displaystyle S = \dfrac{1}{\pi - 1}$

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Altier*

$S_n = \dfrac{1}{\pi} + \dfrac{1}{\pi^2} + \dfrac{1}{\pi^3} + \dfrac{1}{\pi^4} + \ldots \\ \text{This is in GP} \\ a = \dfrac{1}{\pi} \\ r = \dfrac{1}{\pi} \\ S_n = \dfrac{a}{1-r} \\ = \dfrac{\dfrac{1}{\pi}}{1 - \dfrac{1}{\pi}} \\ = \dfrac{\dfrac{1}{\pi}}{\dfrac{\pi - 1}{\pi}} \\ = \dfrac{1}{\pi - 1}$