Power Series Expansion for Logarithm of 1 + x

Theorem
The Newton-Mercator series defines the natural logarithm function as a power series expansion:

valid for all $x \in \R$ such that $-1 < x \le 1$.

Proof
From Sum of Infinite Geometric Sequence, putting $-x$ for $x$:
 * $(1): \quad \ds \sum_{n \mathop = 0}^\infty \paren {-x}^n = \frac 1 {1 + x}$

for $-1 < x < 1$.

From Power Series Converges Uniformly within Radius of Convergence, $(1)$ is uniformly convergent on every closed interval within the interval $\openint {-1} 1$.

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