# Limit of $n^{k/n}$

I want to find the radius of convergence for $$\sum n^k z^n$$, where $$k \in \mathbb{N}$$. Using the root test, I want to find $$lim_{n \rightarrow \infty} \; n^{k/n}$$

I know $$lim_{n \rightarrow \infty} \; n^{1/n} = 1$$, so using properties of convergent sequences, $$lim_{n \rightarrow \infty} \; n^{k/n} = 1^k = 1$$.

First, is this right/Is there a better way to prove this?

Second, this makes no intuitive sense, shouldn't this sequence diverge since $$n^{k/n} \geq n^{1/n}$$?

Yes, it is correct. And, since $$\lim_{n\to\infty}n^{1/n}=1$$, the fact that you always have $$n^{k/n}\geqslant n^{1/n}$$ does not imply at all that $$(n^{k/n})_{n\in\mathbb N}$$ diverges.