This is exercise from Guidorizzi's book Cálculo (In Portuguese).
Show that $\lim_{n\rightarrow \infty}n \big[1-\frac{(n+1)^{n}}{en^{n}}\big]=\frac{1}{2}.$
All I managed to do is rewrite the equation as $n \big[1-\frac{(1+\frac{1}{n})^{n}}{e}\big]$.
What to do from now on?