Leibniz's Formula for Pi/Lemma

Lemma

 * $$\frac 1 {1+t^2} = 1 - t^2 + t^4 - t^6 + \cdots + t^{4n} - \frac {t^{4n + 2}}{1+t^2}$$

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 $$t$$.