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: $|a_n| < \dfrac{K}{n^p} \Rightarrow \displaystyle \sum_{n=N}^\infty |a_n| < K\displaystyle \sum_{n=N}^\infty \dfrac{1}{n^p}< \infty \Rightarrow \displaystyle \sum_{n=1}^\infty |a_n| \text{ converges}\Rightarrow \displaystyle \sum_{n=1}^\infty a_n\text{ converges absolutely}$.