# Convergence of series

1. Jan 23, 2005

### twoflower

Hi all,

I have this series:

$$\sum_{n = 1}^{+\infty} \tan \left( \frac{\pi}{4^{n}} \right) . \sin 2^{n}$$

I have to find out whether it converges or not, but I don't know how should I start. The only idea coming to my mind is to use Abel-Dirichlet's rule for convergence, but I don't know how to prove that sin has limited partial sums. Then I could use the rule I hope.

Or is there any other and more clever way how to prove the convergence?

Thank you.

2. Jan 23, 2005

### vincentchan

Hints
$$\sum_{n = 1}^{+\infty} \tan \left( \frac{\pi}{4^{n}} \right) \cdot \sin 2^{n} < \sum_{n = 1}^{+\infty} \tan \left( \frac{\pi}{4^{n}} \right)$$