An= n[ln(n+1) - lnn]
does it equal e
It doesn't equal e...but you have the right idea.
Note that $\displaystyle a_n=n\left(\ln(n+1)-\ln(n)\right)=n\ln\left(\frac{n+1}{n}\right)=\ln\l eft[\left(1+\frac{1}{n}\right)^n\right]$
Now, $\displaystyle \lim \left(1+\frac{1}{n}\right)^n=e$
So, $\displaystyle \lim a_n=\dots$
Does this make sense?