Leibniz's Formula for Pi/Lemma

Lemma

 * $\displaystyle \frac 1 {1 + t^2} = 1 - t^2 + t^4 - t^6 + \cdots + t^{4 n} - \frac {t^{4 n + 2}}{1 + t^2} = \left({\sum_{k \mathop = 0}^{2 n} \left({-1}\right)^k t^{2k}}\right) - \frac {t^{4 n + 2}}{1 + t^2}$

This holds for all real $t \in \R$.

Proof
From Even Powers are Positive, we have that $t^2 \ge 0$ for all real $t$.

So $- t^2 \le 0$ and so $- t^2 \ne 1$.

So the conditions of Sum of Geometric Progression are satisfied, and so the above argument holds for all real $t$.