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Math Help - convergence of infinite series

  1. #1
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    convergence of infinite series

    Determine whether the following series is convergent (hint: use the comparison test)

    1-(1/2)+(2/3)-(1/3)+(2/4)-(1/4)+....
    ie.
    (2/2)-(1/2)+(2/3)-(1/3)+(2/4)-(1/3)+...

    Am I allowed to turn this into
    (1/2)+(1/2)+(1/4)+..... by summing each pair of terms in the series, or is that only allowed if I know the series converges absolutely?

    If I can do that, then the series is infinitesum{ 1/(n+1) } which equals infinitesum{ 1/n } -1 and since infinitesum{1/n} diverges, so does infinitesum{ 1/n } -1 I would think.

    But I didn't use comparison...
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  2. #2
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    Quote Originally Posted by 234578 View Post
    Determine whether the following series is convergent (hint: use the comparison test)

    1-(1/2)+(2/3)-(1/3)+(2/4)-(1/4)+....
    ie.
    (2/2)-(1/2)+(2/3)-(1/3)+(2/4)-(1/3)+...

    Am I allowed to turn this into
    (1/2)+(1/2)+(1/4)+..... by summing each pair of terms in the series, or is that only allowed if I know the series converges absolutely?


    The second option (i.e., you can't arbitrarily group terms in a general series), but even then I can't see how woould you get what you wrote if you group in pairs...it should be, imo, \frac{1}{2}+\frac{1}{3}+\frac{1}{6}+\ldots...

    Tonio

    If I can do that, then the series is infinitesum{ 1/(n+1) } which equals infinitesum{ 1/n } -1 and since infinitesum{1/n} diverges, so does infinitesum{ 1/n } -1 I would think.

    But I didn't use comparison...
    .
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