Power Series Expansion for Real Arctangent Function

Theorem
The arctangent function has a Taylor series expansion:

which converges for $-1 \le x \le 1$.

Proof
From Sum of Infinite Geometric Progression:
 * $(1): \quad \displaystyle \sum_{n \mathop = 0}^\infty \left({-x^2}\right)^n = \frac 1 {1 + x^2}$

for $-1 < x < 1$.

From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:

For $-1 \le x \le 1$, the sequence $\left \langle \dfrac {x^{2n + 1}} {2n + 1} \right \rangle $ is decreasing and converges to zero.

Therefore the series converges in the given range by the Alternating Series Test.