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The back of my book says it converges, but each test I've tried says the summation diverges. Can someone help me to know if the following converges?

$$\sum_{n=1}^\infty (n+1)\left(\frac{(1+\sin(nπ/6))}{3}\right)^n $$

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Yes, please try Wolfram Alpha (very large number, but converges) - maybe try the Cauchy root test to show. –  Amzoti Sep 21 '12 at 3:17
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Hint: The terms are non-negative, and the $n$-th term is $\le (n+1)\left(\dfrac{2}{3}\right)^n$. Now there should be something nice to compare with, which yields to standard tests.

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Cauchy root test should do the job. $$C = \limsup_n \left( \left \vert (n+1)\left(\frac{(1+\sin(nπ/6))}{3}\right)^n \right \vert \right)^{1/n} = \dfrac23 < 1$$

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