- #1
nameVoid
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Mastering the Series Ratio Test: A Comprehensive Guide
- Thread starter nameVoid
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- Ratio Ratio test Series Test
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Homework Help Overview
The discussion revolves around the application of the ratio test and the comparison test in the context of series convergence, particularly focusing on the relationships between sequences An, an, and bn.
Discussion Character
- Exploratory, Assumption checking, Problem interpretation
Approaches and Questions Raised
- Participants explore the validity of applying the ratio test to sequences and how the results can be used in conjunction with the comparison test. Questions arise regarding the relationships between different sequences and the conditions under which these tests can be applied.
Discussion Status
Some participants provide guidance on the appropriate use of the ratio test and the comparison test, noting the importance of understanding the convergence of the series being compared. There is an ongoing exploration of the implications of negative terms in series and the conditions required for the tests to be valid.
Contextual Notes
Participants highlight constraints related to the presence of negative terms in series and the necessity of knowing the convergence status of the series used for comparison.
- #2
Mark44
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nameVoid said:
What's your question?
- #3
nameVoid
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is it correct to apply the ratio test on An>=an and then use the results in the comparison test where bn=An for bn convergence and bn>an
- #4
Mark44
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I have no idea what you're saying here. You have An, an, and bn. How do these relate to your original series?nameVoid said:
The first test you did was a comparison test, comparing with [itex]\sum 1/n![/itex]. When you use the comparison test, you should already know whether the series you're comparing to converges or diverges. If you are comparing [itex]\sum a_n[/itex] to a convergent series, you have to show that an <= the corresponding term in your convergent series. If you are comparing [itex]\sum a_n[/itex] to a divergent series, you have to show that an >= the corresponding term in the divergent series.
Also, the comparison test and the ratio test have to be used on series without negative terms. Your cosine series has negative terms.
- #5
jtyler05si
- 27
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You could use the ratio test on the original problem; remember when using the ratio test you are taking the limit of the abs value of your function. You could also use the squeeze theorem along with the comparison test.
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