Stirling's Formula/Proof 2/Lemma 2

Theorem
The sequence $\sequence {d_n}$ defined as:
 * $d_n = \map \ln {n!} - \paren {n + \dfrac 1 2} \ln n + n$

is decreasing.

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

Let:
 * $\map f x := \dfrac 1 {2 x} \map \ln {\dfrac {1 + x} {1 - x} } - 1$

for $\size x < 1$.

Then from Lemma 1:


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

Thus $\map f x > 0$ for $\size x < 1$.

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

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

Thus $\sequence {d_n}$ is a decreasing sequence.