Solving this limit $lim_{ntoinfty} left ( frac n {n+1} right)^n$ when applying ratio test on this series..

if I want to check the convergence of the series
$$\sum\frac{n!}{n^n}$$
While solving using Ratio test , I encountered this limit
$$\lim_{n\to\infty} \left( \frac n {n+1} \right)^n$$
is it correct to solve it as following
$$\lim_{n\to\infty} \left( \frac 1 {1+\frac 1 n} \right)^n = \lim_{n\to\infty} \frac 1 { \left( 1+\frac 1 n \right)^n}=e^{-1}$$
(I know that the limit gives here exponential($-1$) but I feel it is wrong to remove the power n from the numerator )

Note:
I solved it also by taking ln to the limit and applying l'hopital , I got exponential(-1) also ,but it is very long .. So I want more shorter way and I do not know whether this shorter way correct or not ..

Answer

It is absolutely correct infact it's always true that:

$$\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}$$