Stirling's Formula/Proof 2/Lemma 1

Lemma
Let $f \left({x}\right)$ be the real function defined on the open interval $\left({-1 \,.\,.\, 1}\right)$ as:
 * $f \left({x}\right) := \dfrac 1 {2 x} \ln \left({\dfrac {1 + x} {1 - x} }\right) - 1$

Then:
 * $\displaystyle f \left({x}\right) = \sum_{k \mathop = 1}^\infty \dfrac {x^{2 n} } {2n + 1}$