Gregory Series

Theorem
For $-\dfrac \pi 4 \le \theta \le \dfrac \pi 4$:
 * $\theta = \tan \theta - \dfrac 1 3 \tan^3 \theta + \dfrac 1 5 \tan^5 \theta - \ldots$

Proof
By the root test the radius of convergence is $-\dfrac \pi 4 \le \theta \le \dfrac \pi 4$.

Also see

 * Leibniz's Formula for Pi