Gregory Series

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

This is called the Gregory series.

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

Also presented as
This series is also presented as:


 * $\tan^{-1} \theta = \theta - \dfrac {\theta^3} 3 + \dfrac {\theta^5} 5 - \ldots$

which is valid for $-1 \le x \le 1$.

Also known as
This is also known as Gregory's series.

Also see

 * Leibniz's Formula for Pi