Complex Series converfence proof

[stihl29]

Homework Statement

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.

Homework Equations

$$\displaystyle \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y},\quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x},$$

The Attempt at a Solution

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

 

[Nino MMP]

$\displaystyle \frac{\partial u_n}{\partial x} = \frac{\partial v_n}{\partial y},\quad\frac{\partial u_n}{\partial y} = -\frac{\partial v_n}{\partial x},$Then take the sum of each CR equation for the infinite sequence. $\displaystyle \sum_{n=0}^\infty\frac{\partial u_n}{\partial x} = \sum_{n=0}^\infty\frac{\partial v_n}{\partial y},\quad\sum_{n=0}^\infty\frac{\partial u_n}{\partial y} = \sum_{n=0}^\infty -\frac{\partial v_n}{\partial x}.$But that doesn't seem right. Can someone help?