Is the sequence $\frac{e^n}{n}$ convergent? Is the sequence $$\frac{e^n}{n}$$ convergent?
I think it is not because $\log n < n$, implying that $\frac{e^n}{n} >1$ and hence the limit does not exist. Which probably also means that the sequence is unbounded.
Am I right? Please correct me if I am wrong.

No it is not. Notice that $$a_{n+1}=\dfrac{e^{n+1}}{n+1}=e\cdot \dfrac{e^n}{n}\cdot \dfrac{n}{n+1}>\dfrac{e}{2}a_n>1.3a_n$$therefore $$a_{n+1}>1.3a_n>(1.3)^2a_{n-1}>\cdots>(1.3)^na_1=e\cdot(1.3)^n$$which is unbounded since $e\cdot(1.3)^n$ is unbounded.
