Convergence of $\sum_{n=1}^\infty{\frac{sin(5n)}{5^n}}$

Question

I have to find if this series is convergent or divergent.

This is the series: $\sum_{n=1}^\infty{\frac{sin(5n)}{5^n}}$

I can't use the Ratio Test, and I don't know what to do with the sine in the numerator. Could someone give me a tip to get started?

Answer 1

You don't need the alternating series test for

$$\sum_{n=1}^\infty{\frac{\sin(5n)}{5^n}} $$

Since $|\sin(5n)| \leq 1$ and noting that $$\left|\frac{\sin(5n)}{5^n}\right|\leq \frac{1}{5^n}$$

it will easily follow that the series converges absolutely.

Answer 2

Since $|\sin \alpha| \leq 1 $ for any $\alpha$, then

$$ \frac{| \sin (5n)| }{|5^n|} \leq \frac{1}{5^n} $$

Now, Use geometric series and compare.