函数项级数一致收敛性的判别与应用

Discrimination and Application of Uniform Convergence of Series of Function Terms

函数项级数一致收敛性的判别问题是分析学的重难点之一。关于函数项级数一致收敛问题在不同题设下判别方法各异,因此函数项级数往往是学生学习数学分析的困难点。为了深入研究函数项级数的一致收敛性,本文对函数项级数一致收敛性的判别法进行全面归纳,并给出每类判别法相对应下的典型例题。通过对比分析,Weierstrass判别法与柯西收敛准则相较于其它方法应用更广泛,故在做题时可优先考虑。

The discrimination of uniform convergence of series of function terms is one of the most difficult problems in mathematical analysis. The discriminant methods of uniform convergence of function term series vary under difficult sets, so the function term series is often the important and difficult point for students to learn mathematical analysis. In order to further study the uniform conver-gence of function term series, this paper summarizes the discriminant method of uniform conver-gence of function term series, and gives the typical examples corresponding to each type of dis-criminant. Through comparative analysis, Weierstrass criterion and Cauchy convergence criterion are more widely used than other methods, so they can be given priority when doing problems.