Power Series Expansion for Half Logarithm of 1 + x over 1 - x

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

Proof
From Power Series Expansion for $\ln \left({1 + x}\right)$:
 * $(1): \quad \displaystyle \ln \left({1 + x}\right) = \sum_{n \mathop = 1}^\infty \left({-1}\right)^{n - 1} \frac {x^n} n$

for $-1 < x \le 1$.

From Power Series Expansion for $\ln \left({1 + x}\right)$: Corollary:
 * $(2): \quad \displaystyle \ln \left({1 - x}\right) = - \sum_{n \mathop = 1}^\infty \frac {x^n} n$

for $-1 < x < 1$.

Then we have:

Hence the result.