Limit of the sequence $\frac{(n+1)^n}{n^{(n+1)}}$

Question

To find the limit of the sequence $$\left\{ \frac{(n+1)^n}{n^{(n+1)}}\right\}$$

It is clear that for $n>3,$

$ (n+1)^n<n^{(n+1)}$

This $\implies 0<\frac{(n+1)^n}{n^{(n+1)}}<1$ for $n>3$

Thus the sequence is bounded and also monotonic which implies this sequence is convergent.

But what is the limit of the sequence and how to find it?

Answer 1

Note that: $$\require{cancel}\frac{(n+1)^n}{n^{n+1}}=\frac{\color{#9999FF}{\cancel{\color{black}{n^n}}}}{\color{#9999FF}{\cancel{\color{black}{n^n}}}\!\!\!\!\cdot\!n}\left(1+\frac{1}{n}\right)^n\!=\frac1n\left(1+\frac1n\right)^n\!\!\underset{n\to\infty}{\longrightarrow}0\!\cdot\!e=0\;.$$ Hence you have: $$\lim_{n\to +\infty} \frac{(n+1)^n}{n^{n+1}}=0\;.$$