Show that the series ((ln n)^q)/n^p from n = 1 to infinity converges if p>1 where q is any real number.
I said that
((ln n)^q)/n^p = ((ln n)^q)/n X 1/(n^(p-1)) < 1/n^(p-1)
From the knowledge of p-series, the original series will converge if p>2. Shouldn't I instead get that p>1?
Also, is there a way to do this question using the limit comparison test? I attempted to use a = ((ln n)^q)/n^p and b = 1/n^p. I got the limit as
(ln n)^q which obviously diverges as n approaches infinity. Where did I make a mistake in this approach?