Difference between Summation of Natural Logarithms and Summation of Harmonic Numbers

Theorem

 * $\displaystyle \sum_{k \mathop = 1}^n H_k - \sum_{k \mathop = 1}^n \map \ln {n!} \approx \gamma n + \dfrac {\ln n} 2 + 0 \cdotp 158$

where:
 * $H_k$ denotes the $k$th harmonic number
 * $n!$ denotes the $n$th factorial
 * $\gamma$ denotes the Euler-Mascheroni constant.

Proof
From Summation over k to n of Natural Logarithm of k:
 * $\displaystyle \sum_{k \mathop = 1}^n \ln k = \map \ln {n!}$

Then:

Then we have that:

Then:

The result follows by ignoring the lower order terms in $n$.