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If you suspect that a series k=1ak converges, explain why you would want to compare the series with a convergent series, using either the comparison test or the limit comparison test.

Short Answer

Expert verified

If the series k=1akis convergent, comparing it to divergent series will yield no results since the behaviour of the series k=1akis dependent on the behaviour of the series k=1bk.

As a result, if the k=1akseries converges, it must be compared to a convergent series.

Step by step solution

01

Step 1. Given information

A series is given ask=1ak

02

Step 2. Verification

The limit comparison test for k=1akand k=1bkare the series having positive terms the the following conditions may apply,

If limkakbk=L, L must be positive number then it may be either converging or diverging.

If limkakbk=0then if k=1bkconverges k=1akconverges

If limkakbk=then if k=1bkdiverges k=1akdiverges

If the series k=1akis convergent, comparing it to divergent series will yield no results since the behaviour of the series k=1akis dependent on the behaviour of the series k=1bk.

As a result, if the k=1akseries converges, it must be compared to a convergent series.

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