Uniform convergence of series: $\sum_{n=1}^{\infty}{2^n\sin\left(\frac{x}{3^n}\right)}$ Task: we should find area of x>=0, that the series is Uniform convergence on this area
$$\sum_{n=1}^{\infty}{2^n\sin\left(\frac{x}{3^n}\right)}$$
There are some ways to proof "Uniform convergence of sum". 
I tried to use Dirichlet and Abel's ways to solve, but they are not suited on this case.
However, I checked answers and got: it is not converges uniformly

$$|\sin{\frac{x}{3^n}}| \leq \frac{|x|}{3^n}$$ so the series converges pointwise on the real line.
The series also converges uniformly in every bounded subset of $\Bbb{R}$ by the $M-$test of Weierstrass.
