Alternating series test for convergence

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Homework Help Overview

The discussion revolves around the convergence of an alternating series, specifically focusing on the application of the alternating series test. Participants are examining the derivation of certain terms and the logic behind the comparisons made in the test.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants question the origin of specific terms (1/2 and n/(n + 1)) and whether demonstrating that an + 1 ≤ an is sufficient. There are discussions about the logic of starting from the result and working backwards, as well as the clarity of different approaches to the problem.

Discussion Status

The conversation is active, with participants exploring different interpretations of the problem and sharing insights about the reasoning behind the steps involved in the test. Some guidance has been offered regarding the comparison of magnitudes and the treatment of signs in the series.

Contextual Notes

Participants are navigating through the nuances of the alternating series test and its requirements, including the need to check the sign of the terms separately from their magnitudes.

Maddie1609
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Homework Statement



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Homework Equations



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The Attempt at a Solution



I don't get how they got what's stated in the above picture. Where does 1/2 and n/(n + 1) come from? Can't you just show that an + 1 ≤ an?
 
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Maddie1609 said:
Where does 1/2 and n/(n + 1) come from? Can't you just show that an + 1 ≤ an?
That's what they do. The logic is easier to follow if you start with the result and reduce this to something true (look at the steps in reverse order), but this direction works as well. The first line is then a clever guess what will be needed later (and the statement is clearly true).
 
mfb said:
That's what they do. The logic is easier to follow if you start with the result and reduce this to something true (look at the steps in reverse order), but this direction works as well. The first line is then a clever guess what will be needed later (and the statement is clearly true).
A lot easier to follow that direction, thanks! The final step has 2n and 2n - 1 instead of (-2)n and (-2)n - 1 which I don't get.
 
The test compares the magnitude of the sequence elements only, powers of -1 don't change the magnitude.
The sign has to be checked separately (easy here).
 
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mfb said:
The test compares the magnitude of the sequence elements only, powers of -1 don't change the magnitude.
The sign has to be checked separately (easy here).
Oh okay! Thank you :smile:
 

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