Comparison Test for Improper IntegralsNow that we’ve seen how to actually compute improper integrals we need to address one moretopic about them. Often we aren’t concerned with the actual value of these integrals. Instead wemight only be interested in whether the integral is convergent or divergent. Also, there will besome integrals that we simply won’t be able to integrate and yet we would still like to know ifthey converge or diverge. To deal with this we’ve got a test for convergence or divergence that we can use to help usanswer the question of convergence for an improper integral.We will give this test only for a sub-case of the infinite interval integral, however versions of thetest exist for the other sub-cases of the infinite interval integrals as well as integrals withdiscontinuous integrands.Comparison TestIf on the intervalthen,1.If converges then so does.2.Ifdiverges then so does.Note that if you think in terms of area the Comparison Test makes a lot of sense. If is larger than then the area under mustalso be larger than the area under .
Example 1 Determine if the following integral is convergent or divergent.