Stirling's approximation

The Stirling's approximation formula:

$${\displaystyle n!\sim \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n,}$$

 

where the sign ~ means that the two quantities are asymptotic: their ratio tends to 1 as ${\displaystyle n}$ tends to infinity.

 

This formula can be used to solve limits of sequences involving factorial.

 

Practice problems:

1. The following problem is asked in CSIR NET 2020(TN-PY)

Find the limit: $\lim_{n \to \infty} \frac{((n+1)(n+2)\cdots(n+n))^{\frac{1}{n}}}{n}$ 

Note: Here we use ${n}^\frac{1}{n} \to 1.$ (Prove it also.)

2. $\lim_{n \to \infty}( \frac{(n!)}{(kn)^n})^{\frac{1}{n}}$ where $k \neq 0$