Cauchy-Hadamard定理中关于“幂级数收敛半径确定”充分性的分析  

Analysis on the Sufficiency of Determining the Convergence Radius of Power Series in Theorem Cauchy-Hadamard

作　　者：李占勇 LI Zhan-yong(School of Mathematics and Statistics,Kashi University,Kashi 844000,Xinjiang,China)

机构地区：[1]喀什大学数学与统计学院,新疆喀什844000

出　　处：《喀什大学学报》2020年第6期17-20,共4页Journal of Kashi University

摘　　要：针对华东师范大学数学系编著的《数学分析(下册)》第三版第十四章第一节Cauchy-Hadamard定理中利用上极限确定幂级数收敛半径的条件"当0<ρ<+∞时,收敛半径R=1/ρ",给出了一个反例说明该条件充分性不足,并通过分析应对幂级数系数集{an}的有界性加以限制,得到了Cauchy-Hadamard定理的最优充分性条件.According to the Cauchy-Hadamard theorem in the first section of Chapter 14 in the third edition of mathematical analysis(Volume II)edited by the Department of mathematics of East China Normal University, the condition of using upper limit to determine the convergence radius of power series" when 0<ρ <+∞,the radius of convergence R=1/ρ",this paper gives a counter example to show that the condition is insufficient,The boundedness of coefficient setan {n√|an|}of power series should be restricted by analysis. Finally, the optimal sufficient conditions of Cauchy-Hadamard theorem are obtained.

关 键 词：Cauchy-Hadamard定理 幂级数收敛半径 充分性 上极限 下极限 

分 类 号：O173.1[理学—数学]