I need to either prove the following or find a counterexample. I really hope you can help, I cannot figure it out.
Let $(a_n)$ be a positive sequence. $$\sum_{n=1}^\infty \frac{a_n}{1+a_n}$$ converges, then $$\sum_{n=1}^\infty a_n$$ converges
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First prove that $a_n\to 0$. – André Nicolas May 01 '16 at 18:42
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Adam Hughes' answer shows the hypothesis that $a_n>0$ is necessary, and gives $a_n = (-1)^n/n$ as a counterexample. – Pedro May 01 '16 at 18:42
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Hint.
When $x$ is positive and $\frac{x}{1+x}$ is sufficiently small, we have $0 < x < 2\frac{x}{1+x} $.
hmakholm left over Monica
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Suppose that such series converges. Note then that since $a_n+1> 1$, we must have that $a_n\to 0$. But then the quotient of $a_n$ with $a_n/(1+a_n)$ tends to $1$.
Pedro
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