CK12 Calculus - Single Variable

ratio of the termsof the two seriesis positive:

.

Usingthe Limit ComparisonTest, because divergesand the limit of the ratio is positive, then

diverges.
Unlikethe ComparisonTest, you do not haveto comparethe termsof both series.You may just makea
ratio of the terms.
Thereis aSimplifiedLimitComparisonTest, whichmay be easierfor you to use.

Theorem( The SimplifiedLimitComparisonTest)Suppose and are serieswithout

negativeterms.If is finiteand positive,then either and both convergeor

and both diverge.

Example 9 Determineif convergesor diverges.
Solution

is a serieswithoutnegativeterms.To applythe SimplifiedLimitComparisonTest, we can

compare with the series , which is a convergent geometric series. Then

. Thus,since converges,then also converges.

ReviewExercises

1. Write an exampleof a nondecreasingsequence.


2. Write an exampleof a sequencethat is not nondecreasing.


3. Suppose{Sn} is a nondecreasingsequencesuchthat for eachM> 0, thereis anN, suchthatSn>M
   for alln>N. Doesthe sequenceconverge?Explain.