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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

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

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  • $\begingroup$ 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 $a_n$? $\endgroup$ – user3704351 Mar 23 '15 at 0:46
  • $\begingroup$ I edited it, and check it out. $\endgroup$ – DeepSea Mar 23 '15 at 0:50
  • $\begingroup$ Thank you, that makes complete sense. I just got confused when you mentioned the comparision test. So it looks like the limit of the supremum portion of the problem is irrelevant? $\endgroup$ – user3704351 Mar 23 '15 at 1:07
  • $\begingroup$ It is relevant because you needed it to get the first inequality going. $\endgroup$ – DeepSea Mar 23 '15 at 1:09

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