Prove absolute convergence for a summation

I need help with this problem. I've been staring at the page blankly tyring to think of ways to solve it. Any hints/solutions would be greatly appreciated.

If $\displaystyle \lim_{n \to \infty}\sup(n^p|a_n|) < \infty$ for some $p > 1$, prove that $\sum_{} a_n$ converges absolutely

It means that $n^p\cdot |a_n| < K$ for some $0 < K < \infty$, and for $n \geq N$ for some $N$ as well. Thus: $|a_n| < \dfrac{K}{n^p} \Rightarrow \displaystyle \sum_{n=N}^\infty |a_n| < K\displaystyle \sum_{n=N}^\infty \dfrac{1}{n^p}< \infty \Rightarrow \displaystyle \sum_{n=1}^\infty |a_n| \text{ converges}\Rightarrow \displaystyle \sum_{n=1}^\infty a_n\text{ converges absolutely}$.
• The inequality above makes sense but how does that help me use the comparison test? What two things to I compare? The inequality above with just $a_n$? – user3704351 Mar 23 '15 at 0:46