Does the sequence converges? I am trying to prove if the sequence 
$a_n=(\root n\of e-1)\cdot n$ is convergent.  I know that the sequences $x_n=(1+1/n)^n$ and $y_n=(1+1/n)^{n+1}$ tends to the same limit which is $e$. Can anyone prove if the above sequence $a_n$ is convergent? and if so, find the limit.
My trial was to write $a_n$ as $a_n=n(e^{1/n}-1)$ and taking $1/n=m$ so that $a_n=\frac{1}{m}(e^m-1)$ and taking the limit $\lim_{x\to 0^+}\frac{e^x-1}{x}$, but I don't know how to continue.
Thanks to every one who solve this for me. 

It is almost finished! Recall the definition of the derivative. Note that the limit you wrote down, apart from being one-sided,  is the derivative of $e^x$ at $x=0$. 
