[tex]\lim _{n\to \infty \:}\left(\frac{n^n}{\left(n+1\right)^n}\right) \\ \frac{n^n}{\left(n+1\right)^n}=\left(\frac{n}{\left(n+1\right)}\right)^n \\ =\lim _{n\to \infty \:}\left(\left(\frac{n}{n+1}\right)^n\right) \\\lim _{n\to \infty }\left(\left(\frac{n}{n+k}\right)^n\right)=\frac{1}{e^k} =\frac{1}{e^1}=\frac{1}{e}[/tex]
