Limit Comparison Test: Does Order Matter?

Thread starter Eats Dirt
Start date Oct 8, 2012
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Comparison Comparison test Limit Test

In summary, the Limit Comparison Test is a mathematical test used to determine the convergence or divergence of a series by comparing it to a known series with similar behavior. It works by taking the limit of the ratio between the terms of the two series and can only be used for series with positive terms. The order of the terms is important and may affect the resulting limit. However, it may not work for series with alternating signs or unclear patterns, in which case other tests may need to be used.

Oct 8, 2012

Eats Dirt

I was just wondering for the limit comparison test does it matter which function is on the top and bottom?

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Oct 9, 2012

MarneMath

Education Advisor

I haven't verified this yet, but doing a rough sketch of the proof, it seems to me that the 'smaller' sequence needs to be the ratio on top.

1. What is the Limit Comparison Test?

The Limit Comparison Test is a mathematical test used to determine the convergence or divergence of a series. It involves comparing a given series to a known series with a similar behavior.

2. How does the Limit Comparison Test work?

The Limit Comparison Test works by taking the limit of the ratio between the terms of the given series and the terms of the known series. If the limit is a positive finite number, then both series either converge or diverge. If the limit is zero or infinity, then the behavior of the two series may differ.

3. What is the importance of the order in the Limit Comparison Test?

The order of the terms in the Limit Comparison Test is crucial as it affects the resulting limit. If the order of the terms is changed, the limit may also change, resulting in a different conclusion about the convergence or divergence of the series.

4. Can the Limit Comparison Test be used for all series?

No, the Limit Comparison Test can only be used for series with positive terms. It also requires the known series to be a convergent series with positive terms.

5. Are there any limitations to the Limit Comparison Test?

Yes, there are some limitations to the Limit Comparison Test. It may not work for series with alternating signs or for series with terms that do not have a clear pattern. In these cases, other tests may need to be used to determine the convergence or divergence of the series.