Study for Uniform Convergence and Power Series

It has been preliminary researched that function series and power series in mathematical analysis course. There are some basic properties and the basic conclusion in the courses. This article is based on the basic theory and properties, for them to make a further in-depth study. First of all, as a necessary tool, it has introduced the two properties of definite integral, it is proved that the continuous function sequence limit problem under the definite integral, then it is defined the sequence of functions on subsets of real number set uniformly Cauchy's concept, basis on them several theorem is proved, it is obtained that results of a series of important properties of function terms. Using of these properties, power series of several important theorems are proved, which is about the important properties of the power series again.