calculus 2 Comparison Test for Improper Integrals -...

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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 TestIfon the intervalthen,1.Ifconverges 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. Ifis larger thanthen the area undermustalso be larger than the area under.
So, if the area under the larger function is finite (i.e.converges) then the area under the smaller function must also be finite (i.e.converges). Likewise, if the area under thesmaller function is infinite (i.e.diverges) thenthe area under the larger function must also be infinite (i.e.diverges).Be careful not to misuse this test. If the smaller function converges there is no reason to believethat the larger will also converge (after all infinity is larger than a finite number…) and if thelarger function diverges there is no reason to believe that the smaller function will also diverge.Let’s work a couple of examples using the comparison test. Note that all we’ll be able to do isdetermine the convergence of the integral. We won’t be able to determine the value of theintegrals and so won’t even bother with that.Example 1Determine if the following integral is convergent or divergent.

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