Uniform Convergence of a series

Hello!

I have the following series:

$\displaystyle \sum _{n=1}^{\infty} 3^{n}\sin\frac{1}{4^{n}x} , x\in(0,\infty)$

I need to prove that it converges, and then check if the convergence is uniformly.

So, I said that:

$\displaystyle 3^{n}\sin\frac{1}{4^{n}x} \le \frac{3^n}{4^{n}x}$

Because for every t : $\displaystyle \sin(t) \le t $

First of all, is this way right? (For every x in range I think that $\displaystyle \sum_{n=1}^\infty \frac{3^n}{4^{n}x}$ converges...)

I also don't have any idea how to check if it converges uniformly (because I can't seem to find the limit-function of this series), and since I can't find a maximum for this series (for the variable x), I can't find a converging series that is larger than my series.

Thank you very much! :)