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Thread: Another series I'm testing for convergence....

  1. #1
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    Another series I'm testing for convergence....

    Sorry to bother with another similar question to the one I was asking earlier, but I have one more series that I'm stuck on...



    I'm pretty sure I did the cancellation right on this one:



    How do you tackle this kind of series when that test fails? I lean towards the comparison test, but I'm having trouble thinking of a comparable series that would appropriately converge or diverge to provide me with meaningful information.
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  2. #2
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by Malaclypse View Post
    Sorry to bother with another similar question to the one I was asking earlier, but I have one more series that I'm stuck on...



    I'm pretty sure I did the cancellation right on this one:



    How do you tackle this kind of series when that test fails? I lean towards the comparison test, but I'm having trouble thinking of a comparable series that would appropriately converge or diverge to provide me with meaningful information.
    Remembering the 'binomial expansion'...

    (1)

    ... and setting in (1) x=1 You find that the series diverges...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
    Last edited by chisigma; Apr 21st 2011 at 07:11 PM. Reason: error in (1)...
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  3. #3
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    Thank you very much. This is an area we have not covered yet, and I will need to study it.
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  4. #4
    MHF Contributor chisigma's Avatar
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    Alternatively You can use the [not very 'popular'...] 'Raabe test' according to which, given a positive terms series...



    ... if for n 'large enough' ...

    (1)

    ... the series converges and if...

    (2)

    ... the series diverges. In your case is...

    (3)

    ... so that the series diverges...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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