sequences and series - How to solve $lim_{nto+infty}n^2left(e-left(1+frac1nright)^nright)$?

I am trying to solve this sequence limit: $$\lim_{n\to+\infty}n^2\left(e-\left(1+\frac1n\right)^n\right)$$ but the only elementary way I found to solve it is to prove that $$ \left(1+\frac1n\right)^n+\frac1n for $n > 3$ and so $$n^2\left(e-\left(1+\frac1n\right)^n\right)>n^2\cdot\frac1n=\frac1n\to+\infty$$ .

However proving the first inequality is not straightforward so I would like to know if there exist a more direct way to solve it.

Answer

I used the fact : $$\lim_{n\to\infty}n\left(e-\left(1+\frac{1}{n}\right)^n\right)=\frac{e}{2}$$

For the large $n$,

$$n^2\left(e-\left(1+\frac{1}{n}\right)^n\right) \thicksim \frac{e}{2}n \longrightarrow\infty \quad \text{where} \quad n \longrightarrow\infty$$