Question

consider the series S(z)=(sum from n=1 to infinity) sin(z)/n^2 (1+cos(piz))
a) prove that this series does not converge uniformly on C.

Answer 1

S(z)=(sum from n=1 to infinity) sin(z)/n^2 (1+cos(piz))
|Fn(x)-F(x)|<e->0
n-> sin(z)/n^w(1+cos(piz)), so at n=0 it goes to infinity.
That means, there is no uniform converging.