Comparison test for sin

Question

I need to use the comparison test for convergence on $\sum_{n=1}^\infty 2^n\sin\frac{(a)}{3^n}.$
I have no idea how to tackle this. Some help/hints would be greatly appreciated. Thank you!

Answer 1

For a hint, note that $|\sin(x)|\leq|x|$ for all $x$.

Answer 2

HINT

Recall that

$$\left| \sin x \right|\le|x| \implies \left|\sin \frac{a}{3^n}\right| \le \frac{|a|}{3^n}$$

then use geometric series.

Answer 3

Use the fact that$$\lim_{n\to\infty}\frac{2^n\sin\left(\frac a{3^n}\right)}{\left(\frac23\right)^n}=a.$$