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

1. Jan 3, 2012

resolvent1

1. The problem statement, all variables and given/known data

I'm trying to show that if

$\sum_{n=1}^\infty x_n$

converges, then so does

$\sum_{n=1}^\infty \frac{x_n}{1-x_n}$

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. Jan 3, 2012

Dick

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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