Do we have some function $f(n)$that converges to the series $\ln n!$ ?

Question

I have tried $f(n) = \int_1^n \ln x dx$, but it fails. The difference is $$\lim_{n\to \infty}(\ln n! - \int_1^n \ln x dx)$$

$\int_1^n \ln x dx = n\ln n -n +1 \\ \ln n! - \int_1^n \ln x dx = \ln n! - n\ln n +n -1 = \ln(\frac{n!e^n}{n^n})-1$

where $\frac{n!e^n}{n^n}\to \infty$, so the limit diverges.

Do we have some function $f(n)$that converges to the series $\ln n!$ ?

Answer 1

By Stirling's approximation

$$ \log n! = \color{red}{\left(n+\frac{1}{2}\right)\log n-n+\log\sqrt{2\pi}}+O\left(\frac{1}{n}\right).\tag{1} $$

Answer 2

Also $\log n! = \sum_{k=1} \log k$ which lies between $\int_{1}^{n} \log xdx$ and $1+\int_{1}^{n}\log x dx$. Now use the squeeze lemma.