Does the series $\sum 2^n \sin(\frac{\pi}{3^n})$ converge? 
Check if $$\sum_{n = 1}^{\infty}2^n \sin\left(\frac{\pi}{3^n}\right)$$ converges.

I tried to solve this by using the ratio  test - I have ended up with the following limit to evaluate: 
$$\lim_{n \to \infty} \left(\frac{2\sin\left(\frac{\pi}{3 \cdot 3^n} \right)}{\sin \left(\frac{\pi}{3^n} \right)} \right)$$
And now - I am stuck and don't know how to proceed with this limit. Any hints?

Same idea as Olivier express in a different way, you know that
$$
\sin\left(x\right)\underset{(0)}{=}x+o\left(x\right)
$$
Hence

$$
2^n\sin\left(\frac{\pi}{3^n}\right) \underset{(+\infty)}{\sim}\pi \left(\frac{2}{3}\right)^n
$$

What can you say about $\displaystyle \sum_{n \geq 0}\left(\frac{2}{3}\right)^n$ ?
