Let $S(n) = \sum_{k = n}^\infty \frac{ln^q k}{k^p}$, $p > 1$
How to determine asymptotic behaviour of the sum $S(n)$, ($n \rightarrow \infty$).
The use of integrals doesn't solve the problem.
Also I used Stolz–Cesàro theorem to solve the problem, but I can't.
What is the first step of solving?
$ S(n) $ ~ ??
Why do you say integrals are of no use? $$ \int_{n+1}^\infty\frac{(\ln x)^q}{x^p}\,dx\le\sum_{k=n}^\infty\frac{(\ln k)^q}{k^p}\le\int_{n}^\infty\frac{(\ln x)^q}{x^p}\,dx. $$ From here you get (integrating by parts) $$ S(n)\sim\frac{(\ln n)^q}{(p-1)\,n^{p-1}}. $$