# Uniform Convergence of a series

• Jul 3rd 2010, 04:52 AM
Uniform Convergence of a series
Hello!

I have the following series:

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

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

So, I said that:

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

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

First of all, is this way right? (For every x in range I think that $\sum_{n=1}^\infty \frac{3^n}{4^{n}x}$ 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! :)
• Jul 3rd 2010, 08:52 AM
roninpro
Your series certainly converges uniformly on any interval $\left[\delta, \infty\right)$ by the Weierstrass M-test, since the term $\frac{3^n}{4^n x}$ would be bounded. However, I am extremely suspicious about including the 0 as the (open) endpoint, because $\sin(\frac{1}{4^nx})$ has a nasty oscillation around 0.

Maybe you can show that the Cauchy Criterion fails?