How to determine if the series $\sum \frac{2+\sin n}{5^n}$ is convergent or divergent?

Question

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!

Answer 1

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 $1 \leq 2 + \sin(n) \leq 3$.