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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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marked as duplicate by David Mitra, 5PM, cardinal, amWhy, Austin Mohr Jan 28 '13 at 3:42

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2 Answers 2

up vote 4 down vote accepted

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 Mahmuud Jan 28 '13 at 2:05

You have ${x\over 1 + x} \le x$ for all $x\ge 0$. That will give you a start.

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Your remark is useful only if the series $\sum a_n$ converges. – Matemáticos Chibchas Jan 28 '13 at 1:59
Yes, but you see the original series Diverges. – Saaqib Mahmuud Jan 28 '13 at 2:06
The series $\sum_n a_n$ and $\sum_n {a_n/(1 + a_n)}$ will converge or diverge together. – ncmathsadist Jan 28 '13 at 2:10

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