Question

Fill in the blanks to complete each of the following theorem statements:

If the improper integral of a function $f$on an interval $I$converges, and if _____ for all $x\in I$where the functions $f$and $g$are defined, then the improper integral of $g\left(x\right)$on $I$also converges.

Short answer

If the improper integral of a function $f$on an interval $I$converges, and if $0\le g\left(x\right)\le f\left(x\right)$for all $x\in I$where the functions $f$and $g$are defined, then the improper integral of $g\left(x\right)$ on $I$ also converges.

Step-by-step solution

Step 1. Given information 

If the improper integral of a function $f$on an interval $I$converges, and if _____ for all $x\in I$where the functions $f$and $g$are defined, then the improper integral of $g\left(x\right)$ on $I$ also converges.

Step 2. Filling in the blanks to complete the theorem statements 

If the improper integral of a function $f$on an interval $I$converges, and if $0\le g\left(x\right)\le f\left(x\right)$for all $x\in I$where the functions $f$and $g$are defined, then the improper integral of $g\left(x\right)$on $I$also converges.

Improper integrals that are smaller than convergent ones will also converge, and improper integrals that are larger than divergent ones will also diverge. This idea makes intuitive sense because a (positive) quantity that is smaller than a finite number will also be finite and a quantity that is larger than an infinite quantity will also be infinite.