A Question about the Alternating Series Test

In summary, the conversation discusses the Alternating Series Test and its convergence when the hypothesis of a decreasing sequence is dropped. The question is how to show the test may fail in this case. One possible solution is to look at a sequence with alternating positive and negative terms, where the odd terms form a convergent sequence and the even terms form a divergent sequence. This sequence will have a limit of 0 and all elements are positive, yet the series is divergent. The suggestion is to use a geometric series for the odd terms and the harmonic series for the even terms.
  • #1
mscudder3
29
0
The definition I am working from is "Let Z=(z(sub n)) be a decreasing sequence of strictly positive numbers with lim(Z)=0. Then the alternating series, Sum(((-1)^n)*Z) is convergent.

My question is how to solve the following:
If the hypothesis that Z is decreasing is dropped, show the Alternating Series Test may fail.

I am aware of a proof utilizing some Z that is also alternating, but this breaks the condition that Z is strictly positive. I am unaware of an such sequence that has a limit of 0, all elements of the series are positive, yet is divergent.

This question is due within 10 hours. Please help!
 
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  • #2
Look at a sequence whose odd terms are a convergent sequence of positive terms, like a geometric series and whose even terms are the terms of the divergent harmonic series {1/n}. The alternating sequence of negative odd terms and positive even terms should diverge. Intuitively this is because what is begin taken away reaches a limit while what is being added doesn't.

It will be a slight nuisance to get the indexing written correctly. I think harmonic terms will be [tex] \frac{1}{ (n/2)+1} [/tex] , for [tex] n = 0,2,4,...[/tex]
 
  • #3
Thanks!

I never really assessed the thought of a limit being reached while another series (running in parallel) continued decreasing. I appreciate the quick input!
 

FAQ: A Question about the Alternating Series Test

What is the Alternating Series Test?

The Alternating Series Test is a mathematical method used to determine if an infinite series, where the terms alternate in sign, converges or diverges. It is based on the concept that if the terms of a series decrease in magnitude and eventually approach zero, the series will converge.

How do you use the Alternating Series Test?

To use the Alternating Series Test, you must first check if the terms of the series alternate in sign. Then, you must check if the terms decrease in magnitude. Finally, you must check if the limit of the terms as n approaches infinity is equal to zero. If all three conditions are met, the series will converge.

What is an example of using the Alternating Series Test?

An example of using the Alternating Series Test is determining the convergence of the infinite series 1 - 1/2 + 1/3 - 1/4 + ... Since the terms alternate in sign and decrease in magnitude, we can use the test. The limit of the terms as n approaches infinity is equal to zero, so the series converges.

Can the Alternating Series Test be used for any type of series?

No, the Alternating Series Test can only be used for infinite series where the terms alternate in sign. It cannot be used for series with non-alternating terms or series with alternating terms that do not decrease in magnitude.

What is the difference between the Alternating Series Test and the Ratio Test?

Both the Alternating Series Test and the Ratio Test are used to determine the convergence or divergence of infinite series. However, the Alternating Series Test is specifically for series with alternating terms, while the Ratio Test can be used for any type of series. Additionally, the Alternating Series Test is based on the behavior of the terms as n approaches infinity, while the Ratio Test is based on the ratio of consecutive terms.

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