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Math Help - Summability and convergence of partial sums

  1. #1
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    Summability and convergence of partial sums

    I'm stuck on a problem here.
    I have the sum of:
    ((-1)^t)/(t+1)

    How do I show that this is not summable? And what can we say about the convergence of its partial sums?
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  2. #2
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    I have the sum of:
    Do you mean \sum_{t=0}^\infty\frac{(-1)^t}{t+1}? This series converges because it is alternating and 1/(t+1) monotonically converges to 0 (Alternating series test). It does not converge absolutely, though, because taking the absolute value of each term makes the harmonic series.
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  3. #3
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    Quote Originally Posted by emakarov View Post
    Do you mean \sum_{t=0}^\infty\frac{(-1)^t}{t+1}? This series converges because it is alternating and 1/(t+1) monotonically converges to 0 (Alternating series test). It does not converge absolutely, though, because taking the absolute value of each term makes the harmonic series.
    Yes. that's what I mean. Haven't learned the keyboard commands yet.
    Well, I'm mainly concerned with the summability. I know the series converges, but the sum is unbounded, and thus cannot be dominated by another summable series for all t.
    But I mainly want to know the best way to show a partial sum sequence that converges.

    Thanks.
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  4. #4
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    Quote Originally Posted by emakarov View Post
    Do you mean \sum_{t=0}^\infty\frac{(-1)^t}{t+1}? This series converges because it is alternating and 1/(t+1) monotonically converges to 0 (Alternating series test). It does not converge absolutely, though, because taking the absolute value of each term makes the harmonic series.
    Just got everything I needed from that Alternating Series Test link though. Thanks a bunch.
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