----- why do you say that clearly the original series converges only for x = n*pi? ----- If x <> 0 mod pi is a rational multiple of 2pi, then {sin(nx)} just repeats the same > 1 distinct values as n —> oo. If x is an irrational multiple of 2pi, then {nx mod 2pi} is dense in the interval [0, 2pi), so the terms sin(nx) do not approach 0 as n —> oo. —Dan