Stirling's Formula/Proof 2/Lemma 2

Theorem
The sequence $\left \langle {d_n} \right \rangle$ defined as:
 * $\displaystyle d_n = \ln \left({n!}\right) - \left({n + \frac 1 2}\right) \ln n + n$

is decreasing.

Proof
The proof strategy is to demonstrate that the sign of $d_n - d_{n+1}$ is positive.

Let:
 * $f \left({x}\right) := \dfrac 1 {2 x} \ln \left({\dfrac {1 + x} {1 - x} }\right) - 1$

for $\left\vert{x}\right\vert < 1$.

Then from Lemma 1:


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

Thus $f \left({x}\right) > 0$ for $\left\vert{x}\right\vert < 1$.

Putting $x = \dfrac 1 {2n + 1}$ it can be seen that $(1)$ is $f \left({\dfrac 1 {2n + 1} }\right)$.

As $-1 < \dfrac 1 {2n + 1} < 1$ it can be seen that $(2)$ can be applied and so:
 * $\forall n \in \N: d_n - d_{n+1} \ge 0$

Thus $\left\langle{d_n}\right\rangle$ is a decreasing sequence.