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> Prove that $f(z)=\sum\_{n=1}^\infty\frac{\sin(nz)}{2^n}$ is analytic on $A=\{z\in\mathbb{C}:|\operatorname{Im}(z)|<\log(2)\}$ 
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I tried expanding $\sin(nz)$ in terms of $e^{inz}$ but that did not help me unless I am doing something wrong. I know [Weierstrass's M-test](https://en.wikipedia.org/wiki/Weierstrass_M-test) comes in to play.