# Math Help - Complex Sequence Limits

1. ## Complex Sequence Limits

I am asked to prove that if $z\in\mathbb{C}$ then $\displaystyle lim_{n\rightarrow\infty} \frac{z^n}{n^n}=0$ and $\displaystyle lim_{n\rightarrow\infty} \frac{z^n}{n!}=0$

So I assume I have to treat these as sequence because $\frac{z^n}{n^n}$ cannot be a function of z since the n is there. I know the definition of the limit but I just get stuck. In $z$ were real I think I would take the log and proceed from there, but that seems considerably harder with the complex logarithm.

Anyway I have:

Let $\epsilon >0$. Now I have to find a $K$ such that when $n>K$, $|\frac{z^n}{n^n}-0|<\epsilon$ which I cannot do.

Can anyone help me... thanks.

2. Well if you want you can use the fact that
z= r exp(it) where r and t are real and then look on the real sequence of $|z|^n/n^n=r^n/n^n$, where presumably you have proven it already in your real analysis course.