对数函数和式的渐近表达式

青青子衿

这个渐近表达式对吗？范围是？
\[ \sum_{k=1}^n\ln\big(\operatorname{round}k\big)\,\succcurlyeq\,\bigg(n-\frac{1}{2}\bigg)\bigg(\ln n-1\bigg)+\dfrac{4}{5} \]

hbghlyj

[*]Log Gamma Function -- from Wolfram MathWorld
[*]Logarithm of the gamma function

hbghlyj

Corollary to Binet's Formula
\[ \lim_{z \to \infty} \abs{\operatorname{Ln} \Gamma(z) - (z - \tfrac 1 2) \operatorname{Ln} z + z - \tfrac 1 2 \ln 2 \pi } \to 0\]