Determine if the sequence an = ln(n)/(n^(1/n)) is convergent.

I have some difficulty with showing that the sequence $$ a_n = \frac{\ln(n)}{n^{1/n}} $$ is divergent. Can anyone help me out with this? Thanks!

1 Answer

First we can observe that $n^{1/n}\to1$. In fact $n^{1/n}=e^{\frac{\ln(n)}{n}}$ and $\frac{\ln(n)}{n}\to0$.

Therefore the denominator tends to $1$ while the numerator tends to $+\infty$.