无穷积分∫_a^(+∞)f(x)dx收敛与被积函数f(x)性质相关的反例

On Counterexamples of Relevance between Infinite Integral ∫_a^(+∞)f(x)dx Convergence and Integrand f(x)

本文主要通过举反例,说明了无穷积分∫a+∞f(x)dx收敛时,被积函数f(x)不一定有极限,不一定连续,甚至不一定有界,即使f(x)非负连续,其极限也不一定是0,从而纠正了学生对该问题似是而非的理解,这种举反例的思想对学生学习数学分析有着极大的启发作用。

The paper elaborated Integrand f(x) might not be limited nor continuous,and might even be bounded when Infinite Integral ∫a+∞f(x)dx is convergent,and the limit of the former might not be 0 even when f( x) is a non-negative continuous function through some counterexamples. Such a teaching approach through counterexamples is greatly helpful for students 'learning of mathematical analysis.