we are given that
(n is greater than or equal to 0)converges normally on E, a subset of C, the complex plane (converging normally means that the series of
for n greater than or equal to 0 converges, where llull=sup lu(z)l (where z is an element of E), is the sup-norm of u. i need to show the following:
1. for each z in E, the series
(n greater than or equal to 0) of complex numbers, converges. denote the sum by v(z)=
(n greater than or equal to 0).
2. the series
(for n greater than or equal to 0) converges to v(z) uniformly on E.
3. if u_n(z) is continuous , then so is v(z).
can someone help me to get started on these problems? i'm pretty lost in this assignment because i am not familiar with visualizing convergence in complex plane. also, the italicized texts mean series, i'm not sure how to insert the sigma symbol in the text. thanks.