Question

If $0\le f\left(x\right)\le g\left(x\right)$for every x$\ge$0 and the improper integral${\int }_0^{\infty }g\left(x\right)dx$converges, then the improper integral ${\int }_0^{\infty }f\left(x\right)dx$converges. The objective is to whether determine the statement is true or false

Short answer

True

Step-by-step solution

To determine given statement is true or false 

The improper integral${\int }_0^{\infty }g\left(x\right)dx$is convergent .Therefore ,

${\int }_0^{\infty }g\left(x\right)dx=A$,where A is finite.

Given that $0\le f\left(x\right)\le g\left(x\right)$.

Therefore,$0\le {\int }_0^{\infty }f\left(x\right)dx\le {\int }_0^{\infty }g\left(x\right)dx$

${\int }_0^{\infty }f\left(x\right)dx\le A$.

Therefore, the improper integral${\int }_0^{\infty }f\left(x\right)dx$converges.

Therefore ,the given statement is true.