Comparison test to determine convergence

In summary, the comparison test is a method used to determine the convergence or divergence of infinite series. It involves comparing a given series to a known benchmark series with established convergence properties. If the series in question is less than or equal to a convergent series, it also converges; if it is greater than or equal to a divergent series, it diverges. This test is useful for series that are difficult to analyze directly.
  • #1
Needassistance0987
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Homework Statement
Hello, is using a comparison test for this question ok? because it looked weird to me somehow
Relevant Equations
if bn>an
then an divergence bn also divergence
1712959380186.png
 
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  • #2
Needassistance0987 said:
Homework Statement: Hello, is using a comparison test for this question ok? because it looked weird to me somehow
Relevant Equations: if bn>an
then an divergence bn also divergence

View attachment 343263
To use the comparison test, you need to compare the general term in the series you're looking at with the general term of a series that is known to diverge. Have you convinced yourself that diverges? If so, then your series also diverges.

You could also use the Nth Term Test for Divergence. If , then the series diverges.

Also, we don't say " divergence" -- we say that diverges or it converges.
 
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  • #3
That will not work. To use the direct comparison test, you have to compare positive numbers. becomes negative. See this. You can use absolute values and compare it with an absolutely convergent series to prove absolute convergence. See this.
 
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FAQ: Comparison test to determine convergence

What is the comparison test for convergence?

The comparison test is a method used to determine the convergence or divergence of infinite series. It involves comparing a given series with a second series whose convergence properties are known. If the series being tested is smaller than a convergent series, or larger than a divergent series, conclusions can be drawn about its convergence or divergence.

How do I apply the comparison test?

To apply the comparison test, you first need to identify a series that you can compare your series to, which has known convergence behavior. For a series Σa_n, find another series Σb_n such that a_n ≤ b_n for all n (for the direct comparison test) or a_n ≥ b_n (for the limit comparison test). If Σb_n converges, then Σa_n converges; if Σb_n diverges, then Σa_n diverges.

What is the difference between the direct comparison test and the limit comparison test?

The direct comparison test requires you to find a series with terms that are directly larger or smaller than the terms of your series. In contrast, the limit comparison test involves taking the limit of the ratio of the terms of the two series as n approaches infinity. If the limit is a positive finite number, both series will either converge or diverge together.

Can the comparison test be used for series with negative terms?

No, the comparison test is not applicable for series with negative terms. The test relies on the assumption that the terms being compared are non-negative. If a series contains negative terms, other convergence tests, such as the alternating series test, may be more appropriate.

What types of series are best suited for the comparison test?

The comparison test is particularly useful for p-series, geometric series, and series that resemble these forms. It is effective when the terms of the series can be compared to a known convergent or divergent series that has a similar structure, allowing for straightforward application of the test.

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