Question

what is the comparison test for improper integrals?

Short answer

If $f\left(x\right)\ge g\left(x\right)\ge 0$on the interval $[0,\infty )$then

1.if${\int }_0^{\infty }f\left(x\right)dx$ converges then so does${\int }_0^{\infty }g\left(x\right)dx$ converges

2.if ${\int }_0^{\infty }g\left(x\right)dx$diverges then so does${\int }_0^{\infty }f\left(x\right)dx$ diverges

Step-by-step solution

The comparison test for improper integral

If $f\left(x\right)\ge g\left(x\right)\ge 0$on the interval $[0,\infty )$then

1.if ${\int }_0^{\infty }f\left(x\right)dx$converges then so does ${\int }_0^{\infty }g\left(x\right)dx$converges

2.if${\int }_0^{\infty }g\left(x\right)dx$diverges then so does${\int }_0^{\infty }f\left(x\right)dx$diverges