What would be the radius of convergence of $\displaystyle e^{sinx}$?
This is just a geometric series, so we need the absolute value of the ratio, $\displaystyle e^{\sin(x)}$ in this case, to be less than one.
So this series converges for all values of x in the set $\displaystyle S=\left\{x:|e^{\sin(x)}|=e^{\sin(x)}<1\right\}$
Note that we have that $\displaystyle 0<e^x<1$ iff $\displaystyle x<0$. So in our case we need $\displaystyle \sin(x)<0$ or that $\displaystyle \pi<x<2\pi$ or $\displaystyle 3\pi<x<4\pi$ or in general $\displaystyle (2n-1)\pi<x<2n\pi~~n\in\mathbb{Z}$