$(n^{k})$ diverges whenever k is positive.

Question

Is it true that the sequence $(n^{k})$ diverges for $k$ a fixed positive?

You should be able to prove this simply by showing that the sequence is not bounded, correct?

I just want to make sure that I'm not missing something.

Answer 1

Yes, and you can show it's not bounded, given a positive number $M$, any $n>M^{1/k}$ will give an output higher than $M$, since $x^k$ is monotonically increasing when $k>0$

(this is assuming k is fixed and it's a limit as n goes to infinity)