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Positive series problem
If $a_n > 0$ for each $n = 1, 2, 3, \cdots$ such that the series $\sum_{n=1}^{\infty} a_n$ diverges, then how to determine if the series $$\sum_{n=1}^{\infty} \frac{a_n}{1+a_n}$$ converges or diverges?
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If $a_n$ does not tend to $0$, then clearly $\dfrac{a_n}{1+a_n}$ also doesn't tend to $0$ and hence the series diverges.
If $a_n \to 0$, then we can lower-bound $\dfrac{a_n}{1+a_n}$ by $c a_n$, where $c$ is a constant. (I will let you fill in the details.) Hence, again by comparison test $a_n$ diverges.
Equivalently, you could directly use the limit comparison test.
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I've managed to fill in the details. So nice of you! – Saaqib Mahmood Jan 28 '13 at 02:05
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You have ${x\over 1 + x} \le x$ for all $x\ge 0$. That will give you a start.
ncmathsadist
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Your remark is useful only if the series $\sum a_n$ converges. – Matemáticos Chibchas Jan 28 '13 at 01:59
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The series $\sum_n a_n$ and $\sum_n {a_n/(1 + a_n)}$ will converge or diverge together. – ncmathsadist Jan 28 '13 at 02:10