By comparison with $\sum |x|^n$, our series converges absolutely if $|x|<1$. Let's see why things go bad for most $|x|\ge 1$. Except when $x$ is of the form $k\pi$, the terms do not have limit $0$.
To do this, you will have to show that except in the case when $x$ is an integer multiple of $\pi$, we can find a positive $\alpha$ such that infinitely many integers $n$, $|\sin(nx)|>\alpha$. Hint: Suppose that by bad luck $\sin(nx)$ is awfully close to $0$. Show that $\sin((n+1)x)$ isn't.
Remark: I am not sure about the use of the term convergence radius. With power series, we have divergence if $|x|$ is greater than the convergence radius. Here we mostly have divergence, but the points $k\pi$ are exceptional.