Infinite series (ln(n))^998/n^(7/8) diverges

Question

Can someone please explain why the infinite series (ln(n))⁹⁹⁸/n^(7/8) diverges and doesn't converge? I was looking at the relative growth rates and noticed that n^q (with q being a number greater than zero) grows faster than (ln(n))^p (with p being a number greater than zero) so I thought the series would converge, but apparently it diverges. I do not know how to prove it, I was thinking it has something to do with the direct comparison test/basic comparison test, but I wouldn't know what to compare it to.

Answer 1

1) p and q must be greater than 1 for your observation to be true

2) in this case q < 1 and division by a proper fraction increases the quotient.