[math-fun] Where does this series converge?

I recently tried to make a plot of (*) y = Sum_{1 <= n < oo} sin(nx) for some small values of infinity, and it distinctly hovered around the apparent value Sum_{1 <= n < oo} Im( exp(inx)) = Im(Sum_{1 <= n < oo} exp(inx)) =(???) Im( exp(ix) / (1 - exp(ix)) ) = sin(x) / (2 - 2*cos(x)). Clearly the original series converges only for x = n*pi for some integer n. Can this function f(x) = sin(x) / (2 - 2*cos(x)) be the *Cesaro sum* of the original series (*) ??? Or at least for certain values of x ??? —Dan