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Consider $$\sum_{n=1}^\infty (3n + \cos(n))^{-n}$$ Is this series either divergent, absolutely convergent, or conditionally convergent?

I've attempted to just start with one type of divergence/convergence (or lack thereof), but didn't get far on this specific question and, generally, find these types of questions very time-consuming (is there a standard go-to method? I usually work on divergence or absolute convergence first)

Thanks for any help.

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    $\begingroup$ For any $n\geq 2$, $$0\leq (3n+\cos n)^{-n}\leq (3n-1)^{-n} \leq \frac{1}{(3n-1)^2}\leq\frac{1}{n^2}.$$ $\endgroup$ – Jack D'Aurizio Jun 13 '15 at 15:10
  • $\begingroup$ @JackD'Aurizio Just out of curiosity, why didn't you post this as an answer? $\endgroup$ – qmd Jun 13 '15 at 15:12
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    $\begingroup$ @SuH: it feels like stealing from kids :D $\endgroup$ – Jack D'Aurizio Jun 13 '15 at 15:29
  • $\begingroup$ @JackD'Aurizio You mean person ;) $\endgroup$ – qmd Jun 13 '15 at 15:30

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