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Math Help - Partial sum convergence

  1. #1
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    Partial sum convergence

    Hey all, I'm having a bit of trouble on this one:

    Suppose \sum^{\infty}_{1} a_{k} is a series with partial sums S_{n} = \frac{1}{2n +1}, n \in N

    a) Explain why \sum^{\infty}_{1} a_{k} is convergent and find the sum of the series.

    b) Find a_{k}, k = 1, 2, ...


    Some of the scratch work I've been doing:
    S_{1} = \frac{1}{3} = a_{1}
    S_{2} = \frac{1}{5} = a_{1} + a_{2}
    S_{3} = \frac{1}{7} = a_{1} + a_{2} + a_{3} ...

    a_{1} = \frac{1}{3}
    a_{2} = -\frac{2}{15}
    a_{3} = -\frac{2}{35}

    \sum^{\infty}_{1} a_{k} = \frac{1}{3} - 2(\frac{1}{15} + \frac{1}{35} + \frac{1}{63} + ... + \frac{1}{(2n+1)(2n+3)} + ... )

    Something is obviously not jumping out at me. Any help would be appreciated.
    Last edited by HeirToPendragon; June 9th 2009 at 07:39 PM.
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  2. #2
    Super Member Random Variable's Avatar
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    The limit of the partial sums of a convergent series is the sum of the series.
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  3. #3
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    Yeah I knew that much, but something in the back of my head was trying to tell me that that was way to easy.

    Sn -> 0 right? Please tell me I'm wrong and there is actually some work to do here.
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  4. #4
    Super Member Random Variable's Avatar
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    Unless it's a trick question, there doesn't seem to be much to say. I guess you could try and explain why a series converges iff its sequence of partial sums converges.
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  5. #5
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    This is aggravatingly dumb. I'll ask the professor tomorrow I guess.

    Alright then, so what is a_{k}, k = 1,2... ?
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  6. #6
    MHF Contributor chisigma's Avatar
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    The solution in implicit in definition of 'partial sum' ...

    S_{n} = \sum_{k=1}^{n} a_{k} (1)

    From (1) we derive immediately...

    a_{n} = S_{n} - S_{n-1} = \frac{1}{2n+1} - \frac{1}{2n-1} = \frac{2}{1-4n^{2}} (2)

    The series of the a_{n} is convergent and its sum is...

    S=\lim_{n \rightarrow \infty} S_{n} = 0 (3)

    Kind regards

    \chi \sigma
    Last edited by chisigma; June 9th 2009 at 10:59 PM.
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