Series Absolutely Convergent, xn/(1+xn^2) absolutely convergent

**jsmith90210** · November 6th 2011, 12:04 PM

Given the infinite series xn is absolutely convergent, show that the infinite series xn/(1+xn) is absolutely convergent.

I am completely stumped, I have tried using properties of the limit test and the comparison test to show this, but I can't seem to make any progress,

**girdav** · November 6th 2011, 12:36 PM

Since the series $\sum_n |x_n|$ is convergent, we have in particular that $|1+x_n|\geq \frac 12$ for $n$ large enough. Hence we have $0\leq \left|\frac{x_n}{1+x_n}\right|\leq 2|x_n|$ , and you can conclude.

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