Stirling approximation for factorial

Prove that for large $n$ the following equation:

$$\ln n ! \approx n \ln n - n $$

holds.

Solution



We note that
$$\begin{aligned}
\ln n! &= \ln \left ( 1\cdot 2 \cdot 3 \cdots n \right ) \\
 &= \ln \left [ \prod_{i=1}^{n}i \right ]\\
 &= \sum_{i=1}^{n}\ln i\\
 &\approx  \int_{1}^{n}\ln x \, {\rm d}x \\
 &=\left [ x \ln x -x \right ]_1^n \\
 &= n \ln n -n +1 \\
 &\approx n \ln n -n
\end{aligned}$$

which is what we wanted.

The above result is a powerful one when needed to estimate series or integrals involving logs.