关于一道高等数学竞赛试题的探索与拓广

Exploration and Extension on an Examination Question of Higher Mathematics Competition

给出了2004年浙江省大学生高等数学竞赛一题得分率较低的压轴题（判断级数∑n=1↑∞ 1/n√（n!）^α的敛散性，其中α〉0为常数）的五种不同的解法，建立了它的如下的拓广结果：当α〉1且正项级数∑i=1↑∞ 1/ai^α收敛时，级数∑n=1↑∞ 1/n√（∏i=1 ↑nai）^α收敛；当0〈α≤1，0〈m〈1/αi〈M且正项级数∑i=1↑∞ 1/ai发散时，级数∑n=1↑∞ 1/n√（∏i=1↑nai）^α发散，其中m和M为两正常数．

This paper gives five different methods of solving a lower scoring examination question （Judge the convergence or divergence of series∑n=1↑∞ 1/n√（n!）^α , where α〉0 and a is constant） on higher mathematics competition of Zhejiang Province in 2004. Then we establish extended results as follow： series of positive terms ∑n=1↑∞ 1/n√（∏i=1 ↑nai）^α divergences if series∑i=1↑∞ 1/ai divergences and 0〈α≤1,0〈m〈1/αi〈M ,but it convergenees if series ∑i=1↑∞ 1/ai^α convergences and α〉1, where m and M are two positive constants.