Is $\frac{1}{(\log(n))^{n^p}}$ convergent or divergent? My solution:
Use ratio test $\lim_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim_{n \to \infty}|\frac{(\log(n))^{(n)^{p}}}{(\log(n+1))^{(n+1)^{p}}}|$
Since we know that as n goes to infinity $\log(n+1)>\log(n)$ and $(n+1)^{p}>n^{p}$, so $$\lim_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|<1$$ and it should converge.
Is that correct?
Your solution is incorrect as you flipped your ratio upside down. Your RHS in the limit is the inverse of the LHS. Also listen to what pirate is telling you. You must also evaluate the limit.
EDIT: Here is a hint to evaluating the limit. $$\lim_{n \to \infty} \frac{\log(n)^{n^p}}{\log(n+1)^{(n+1)^p}} = \lim_{n \to \infty} e^{{\log(\log(n)^{n^p}) -\log(\log(n+1)^{(n+1)^p}})}$$ Now try to use some more logarithm properties to find for which values of $p$ this ratio is less than 1.