反常积分敛散性的根值判别法

The Root Test for Convergence of Improper Integral

由于积分与级数在理论上是统一的,因此有关正项级数的根式判别法可被推广以判别无穷限积分和瑕积分的敛散性.设f(x)是[a,+∞)上的非负函数,li mx→+∞xf(x)=ρ,则当ρ<1时,反常积分∫a+∞f(x)dx收敛,而当ρ>1时,反常积分∫a+∞f(x)dx发散;设f(x)是(a,b]上的非负函数,a为瑕点,xli→ma+(f(x))x-a=ρ,则当ρ<1时,反常积分∫abf(x)dx收敛,而当ρ>1时,反常积分∫baf(x)dx发散.

the integral and series are unified in theory, so the Root Test can be used to examine the improper integral∫a^＋∞f（x）dx for convergence or divergence. Assuming f（x） is non-negative in [a, ＋∞）, and limx→＋∞x√f（x）=ρ, if ρ〈1.the improper integral is convergence,while ρ〉1.the integral is divergence ; assuming f（x） is non-negative in （a,b], a is the spot,limx→a^＋（f（x））^x-a=ρ .if ρ〈1.∫a^bf（x）dx is convergence, while ρ〉1, the integral is divergence.