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I need to see if this series $\sum_{n=1}^\infty\limits \left(\frac {1+\cos(n)}3 \right)^n$ either converges or diverges. I was thinking that because the inside terms are going to fluctuate between $(0,\frac 23)$, the inside is never negative, so it's going to diverge because a sum of positive numbers raised to a power are strictly increasing? Is my logic correct here and/or if there is a theorem that strengthens my argument, it would be appreciated.

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2 Answers 2

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Hint: $$ \sum_{n=1}^\infty \left|\frac{1+\cos(n)}3 \right|^n \leq\sum_{n=1}^\infty \left(\frac23\right)^n<\infty, $$ hence the series converges absolutely, hence converges.

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    $\begingroup$ This isn't a hint $\endgroup$
    – Rick
    Nov 4, 2015 at 1:11
  • $\begingroup$ @Goobys Too detailed? But, as we may deduce fro the question, OP will have enough work with the remaining details. $\endgroup$
    – Kola B.
    Nov 4, 2015 at 1:21
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    $\begingroup$ What I was referring to was the fact that you said "Hint", yet still gave the entirety of the solution in your answer. A hint is not supposed to give the whole answer. $\endgroup$
    – Rick
    Nov 4, 2015 at 1:45
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The series converges. As you noted $1 + \cos(n) \leq 2$ so we have:

$\Sigma_{n=1}^{\infty} \frac{1+\cos(n)}{3} \leq \Sigma_{n=1}^{\infty} (2/3)^n = 2$ by the geometric series forumla.

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