A Question about the Alternating Series Testby mscudder3 Tags: alternating series, convergence, limit 

#1
Apr711, 11:04 PM

P: 29

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! 



#2
Apr811, 12:33 AM

Sci Advisor
P: 3,166

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
Apr811, 01:19 AM

P: 29

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! 


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