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

Theorem
where $H_n$ denotes the $n$th harmonic number:
 * $H_n = \displaystyle \sum_{r \mathop = 1}^n \dfrac 1 r = 1 + \dfrac 1 2 + \dfrac 1 3 \cdots + \dfrac 1 r$

valid for all $x \in \R$ such that $\left\lvert{x}\right\rvert < 1$.