无穷限积分integral from n=a to +∞ (f(x)dx)中被积函数f(x)的渐近分析  

The Limit Analysis on Integrated Function f(x) of Abnormal integral from n=a to +∞(f(x)dx)

作　　者：熊骏[1] 曾鹏 

机构地区：[1]长江大学信息与数学学院,湖北荆州434023 [2]沙市第九中学,湖北荆州434000

出　　处：《长江大学学报（自然科学版）》2011年第9期4-5,13,共2页Journal of Yangtze University(Natural Science Edition)

摘　　要：讨论了收敛的无穷限积分integral from n=a to +∞ (f(x)dx)中被积函数limx→+∞f(x)=0的充分条件以及在一些条件下limf(x)=0的速度,得到了如下结果:若无穷限积分integral from n=a to +∞ (f(x)dx)收敛且极限limx→+∞f(x)存在,则limx→+∞f(x)=0;若无穷限积分integral from n=a to +∞ (f(x)dx)收敛,且f(x)在[a,+∞)上一致连续,则limx→+∞f(x)=0;若无穷限积分∫+∞af(x)dx收敛且f(x)在[a,+∞)上单调,则limx→+∞xf(x)=0,即f(x)=o1(x),x→+∞;若无穷限积分integral from n=a to +∞ (f(x)dx)收敛(p>0),且f(x)在[a,+∞)上单调,则limxp+1 f(x)=0。In this paper,the sufficient conditions about limx→+∞f(x)=0 and the limits of functions in abnormal integration are discussed,some results are obtained:if integral from n=a to +∞(f(x)dx) converges to a limit and limx→+∞f(x) exists,then limx→+∞f(x)=0;when integral from n=a to +∞(f(x)dx) converges to a limit,and f(x) is continuous uniformly on [a,+∞),then limx→+∞xf(x)=0,says f(x)=o(1/x),x→+∞;if integral from n=a to +∞(f(x)dx) converges to a limit(p>0),and f(x) is monotonous on [a,+∞),then limx→+∞xp+1f(x)=0.

关 键 词：无穷限积分 被积函数 渐近分析 

分 类 号：O175.1[理学—数学]