CK12 Calculus - Single Variable

Then.

Thus,.

Sincethe integralis finite,the series convergesby the IntegralTest.

LimitComparisonTest, SimplifiedLimitComparisonTest
Anothertest we can use to determineconvergenceof serieswithoutnegativetermsis theLimitComparison
Test. It is easierto use than the ComparisonTest.

Theorem(TheLimitComparisonTest)Suppose is a serieswithoutnegativeterms.Thenone of

the followingwill hold.1. If is a convergentserieswithoutnegativetermsand is finite,

then converges.2. If is a divergentserieswithoutnegativetermsand is positive,

then diverges.

The LimitComparisonTest saysto makea ratio of the termsof two seriesand computethe limit.This test
is mostusefulfor serieswith rationalexpressions.

Example 8 Determineif convergesor diverges.
Solution

Just as with rationalfunctions,the behaviorof the series whenkgoesto infinitybehaves

like the serieswith only the highestpowersofkin the numeratorand denominator:. We will use

the series to applythe LimitComparisonTest. First,whenwe simplifythe series , we get

the series. This is a harmonicseriesbecause and the multiplier doesnot

affect the convergenceor divergence.Thus, diverges.So, we will next checkthat the limit of the