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Absolute and uniform convergence of $\sum _{n=1}^{n=\infty }2^{n}\sin \frac {1} {3^{n}z}$
I need to show that $\sum_{k=1}^\infty 2^k \sin\left(\frac{1}{3^k x}\right)$ is aboslutely convergent for $x$ nonzero but not uniformly convergent on $(\epsilon, \infty)$. I have the absolute part but for the uniform I'm trying to show the tail $\sum_{k=n}^m 2^k \sin\left(\frac{1}{3^k x}\right)$ is less than $\epsilon$ for $n$ sufficiently large. I can't see to create the bound.