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I have:

the series $\sum_{n=0}^{\infty} \frac{(2+\sin n)}{5^n} $.

I have split this up into $\sum_{n=0}^{\infty} \frac{2}{5^n} + \sum_{n=0}^{\infty} \frac{\sin n}{5^n}$. I know the first part is convergent by using geometric, but I am not sure how to approach the second part. Please help, thank you!

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    Splitting turns out to be harmless, since one can prove that the second series converges. However, it is not a good idea. Definitely easier is comparison with $\sum_1^\infty \frac{3}{5^n}$.2012-11-07

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There are many ways to show convergence/divergence of a series. Try the ratio test, the root test, the comparison test, or the integral test just to name a few. See here or here for a more thorough listing of tests for convergence.

In this particular case, the comparison test will be best. It may help to note that $-1 \leq \sin(n) \leq 1$ so that $1 \leq 2 + \sin(n) \leq 3$.

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    You could also find the limit of the second part by noting that $\sin n =\Im{e^{i n}}$, to obtain that $\Im{\sum_0^{\infty}\frac{e^{i n}}{5^n}}=\frac{5 \sin 1}{26-10 \cos 1}$. The first sum is easy ...2014-12-18