Limit Comparison Test for Series

In summary, when choosing a bn to compare to a given an, it is important to select a value for bn that you already know converges. Additionally, you should aim for a bn that makes it convenient to evaluate the limit of an/bn, while also ensuring that it meets the criteria for the comparison test.
  • #1
mohabitar
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Just a general question, but I find it hard to come up with a b[n] to compare to a[n]. When the book does it, they come up with stuff to compare to a[n] that I would have never thought of. Is there any criteria, things to look for, etc., for coming up with a b[n] to compare to a[n]?
 
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  • #2
Obviously you need to use bn for which you already know the convergence. You also want to pick bn for which the limit of an/bn is convenient and within the criteria of the comparison test.
 

What is the Limit Comparison Test for Series?

The Limit Comparison Test is a method used to determine the convergence or divergence of a given series by comparing it to a known series with known convergence properties.

How does the Limit Comparison Test work?

The test compares the ratio of the terms of the given series to the terms of the known series as the number of terms approaches infinity. If the resulting limit is a positive finite number, then both series have the same convergence properties. If the limit is 0 or infinity, then the convergence properties of the two series may differ.

When should the Limit Comparison Test be used?

The Limit Comparison Test is best used when the given series is difficult to directly compare to known series using other convergence tests, such as the Comparison Test or the Ratio Test.

What is the difference between the Limit Comparison Test and the Comparison Test?

While both tests involve comparing a given series to a known series, the Limit Comparison Test compares the ratio of the terms of the two series, while the Comparison Test compares the actual terms of the two series. The Limit Comparison Test can be easier to use for series with more complex terms.

What happens if the Limit Comparison Test is inconclusive?

If the resulting limit is 1, the test is inconclusive and other convergence tests should be used to determine the convergence or divergence of the given series.

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