Limit with criteria $lim_{n to infty}n cdot left [ frac1e left (1+frac{1}{n+1} right )^{n+1}-1 right ]$

$$\lim_{n \to \infty}n \cdot \left [ \frac{\left (1+\frac{1}{n+1} \right )^{n+1}}{e}-1 \right ]$$

I was trying to calculate a limit that drove me to this case of Raabe-Duhamel's test, but I don't know how to finish it. Please give me a hint or a piece of advise.

I cannot use any of the solution below, but they are clear and good. I'm trying to prove it using squeeze theorem like this:

$$\lim_{n \to \infty}n \cdot \left [ \frac{\left (1+\frac{1}{n+1} \right )^{n+1}}{e}-1 \right ]=\frac{-1}{e} \cdot\lim_{n \to \infty}n \cdot \left [e- \left (1+\frac{1}{n+1} \right )^{n+1} \right ]$$
I found this:
$$\frac{e}{2n+2}
Is this true? How can I prove this? Thanks for the answers.