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I need help,

If $\forall n\in \mathbb{N}, x_n> 0 $ and $ \sum_{n=1}^{\infty} x_n$ is divergent then $ \sum_{n=1}^{\infty} \frac{x_n}{1+x_n}$ divergent?

thanks for the help

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If $\lim_{n\to\infty} x_n=0$, then for large enough $n$ we have $x_n\lt 1$, and therefore $\frac{x_n}{1+x_n} \gt \frac{x_n}{2}$. It follows by comparison that $\sum \frac{x_n}{1+x_n}$ diverges.

If it is not the case that $\lim_{n\to\infty} x_n=0$, then $\frac{x_n}{1+x_n}$ does not have limit $0$, and therefore $\sum \frac{x_n}{1+x_n}$ diverges.

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  • $\begingroup$ I'm sure that I have seen this before. Even so, yours is a nice solution. $\endgroup$ – marty cohen Apr 13 '16 at 4:44
  • $\begingroup$ @martycohen: Undoubtedly the solution is not new, and it is probably not new to MSE. As time passes, there will be fewer and fewer new solutions. $\endgroup$ – André Nicolas Apr 13 '16 at 4:52
  • $\begingroup$ But there will always be some. $\endgroup$ – marty cohen Apr 13 '16 at 5:12

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