# Complex Series converfence proof

1. Jan 19, 2010

### stihl29

1. The problem statement, all variables and given/known data
Suppose we already know series $$u(z) = \displaystyle\sum_{n=0}^\infty u_n(z)$$is uniformly convergent in the entire complex plain and we can perform term by term integration and differentation each term $$u_n(z)$$ in the analyitic function. use cauchy-riemann equations to show that the sum of u(z) is analytic in the entire complex plain.

2. Relevant equations
$$\displaystyle \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y},\quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x},$$

3. The attempt at a solution
My only guess at a solution would be to use the CR equations for each term in the sequence. ??