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Math Help - Alternating Series Test

  1. #1
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    Alternating Series Test

    * I hope its alright that I reposted this. I realized that Calculus might not be the best category for this. *

    Show by example that the hypothesis b1>=b2>=...bn>=0 cannot be replaced by bk>=0 and limit k-->infinity =0

    hint: use |ab|< 1/2(a2+b2)

    I've found an example: bk=(1/k2 + 1/k) which satisfies the limit going to zer and all terms being positive, which diverges. I'm getting stuck with a rigorous proof of this, I thought about using the Dirichlet Test to prove this, but I'm getting hung up I think. Any help?
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  2. #2
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    Re: Alternating Series Test

    I'm afraid your counterexample is actually convergent. That is \sum^\infty_{k=1}(-1)^{k+1}(1/k^2+1/k) is the sum of two convergent alternating series. I really don't see what example the hint is implying. However, here's an example and a few more comments.

    Alternating Series Test-mhfalternatingseries.png
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