Determining the convergence and divergence So I have a problem of 
$$a_n = \left(1 + \frac{2}{n}\right)^n$$
I need to determine whether it is diverging or converging and find the limit if it is converging I found an answer on symbol lab of $e^2$ but I do not know how they got that type of answer

Well, once you know the series converges, that is actually one of the definitions of $e^2.$ Here is another way. 
Consider $\ln(a_n)=n\ln\left(1+\frac{2}{n}\right).$
Now we look at the form
$$\frac{\ln(1+2/n)}{1/n}=\ln(a_n).$$
We see that 
$$\frac{(\ln(1+2/n))'}{(1/n)}=\frac{\frac{1}{1+2/n}}{-1/n^2}\cdot\frac{-2}{n^2}=\frac{2}{1+2/n}\to 2.$$
So by L'Hopital's 
$$\ln(a_n)\to 2.$$
By continuity 
$$e^{\ln(a_n)}=a_n\to e^2.$$ 
