I'm being asked to show that a given series (of rational functions) converges uniformly on a given disc, and then and asked to use this fact to show that integrating its limit function (i.e. a summation of rational functions) along a given contour results in a given series.
In the solution provided, the arugment is that each of the rational function is continuous, thus the uniform convergence implies that the limit function is continuous too.
I don't understand the need to invoke the uniform-convergence-implies-continuous-limit theorem. Isn't a series of rational functions already automatically continuous?