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Homework Help: Sum of (x_n/(1-x_n)) converges if the sum of x_n converges

  1. Jan 3, 2012 #1
    1. The problem statement, all variables and given/known data

    I'm trying to show that if

    [itex] \sum_{n=1}^\infty x_n [/itex]

    converges, then so does

    [itex] \sum_{n=1}^\infty \frac{x_n}{1-x_n} [/itex]

    2. Relevant equations

    Unknown. Possibly using the limit of the sum of x_n to bound the partial sums of x_n/(1-x_n)?

    3. The attempt at a solution

    My first thought was to try to combine the Cauchy Schwartz inequality with convergence of the sum of 1/(1-x_n) - but this doesn't help since 1/(1-x_n) can diverge. So I don't know how to proceed with this one.
  2. jcsd
  3. Jan 3, 2012 #2


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    Science Advisor
    Homework Helper

    If the series x_n converges, then limit n->infinity x_n=0. Use that to put a bound on the size of the denominator 1-x_n.
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