Stirling's Formula/Proof 2/Lemma 1

Lemma
Let $\map f x$ be the real function defined on the open interval $\openint {-1} 1$ as:
 * $\map f x := \dfrac 1 {2 x} \map \ln {\dfrac {1 + x} {1 - x} } - 1$

Then:
 * $\ds \map f x = \sum_{k \mathop = 1}^\infty \dfrac {x^{2 n} } {2 n + 1}$