无穷积分的收敛与被积函数极限之间关系的探讨

On the relation between convergence of infinite integral and the limit of the integrand

文章讨论无穷积分∫_a^(+∞)f(x)dx的被积函数f(x)当x→+∞时的极限情况。方法:利用函数f(x)在[a,+∞)上一致连续的一些性质、结论和一些新颖的实例。结果:给出了无穷积分∫_a^(+∞)f(x)dx的被积函数极限limf(x)x→+∞的一些条件及其证明。结论:若无穷积分∫_a^(+∞)f(x)dx收敛时被积函数极限为零,必须附加一定的条件才能成立,这与数项级数和函数项级数收敛时一般项趋于零是有差别的。

Objective:To discuss the limit case of integrand f(x)of infinite integral from n=a to (+∞)f (x)DX when x →+∞. Method :use the consistent continuous nature and conclusions and some novel instances of function f(x)on [a,+ ∞) .Results:Given some conditions and its proof when the limit of integrand f(x)of infinite integral from n=a to (+ ∞)f (x)DX is zero when x → +∞. Conclusion:the limit of integrand f(x)is zero when infinite integral from n=a to (+ ∞)f (x)DX is convergent when x → +∞ must be attached to certain conditions.this and several series and the function of the series converges to zero when a general is not consistent.