Uniform convergence of series: $\sum_{n=1}^{\infty}{2^n\sin\left(\frac{x}{3^n}\right)}$

Question

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

Answer 1

$$|\sin{\frac{x}{3^n}}| \leq \frac{|x|}{3^n}$$ 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.