Convergent or divergent series examples

Suppose $\sum a_n$ is convergent. Is $\sum {{a_n} \over {1+|a_n|}}$ convergent or divergent?

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Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. If this is homework, please add the homework tag; people will still help, so don't worry. – Did Sep 10 '12 at 6:15
First consider the simple case that all $a_n\ge0$ and compare $a_n$ with $\frac{a_n}{1+|a_n|}$. – Hagen von Eitzen Sep 10 '12 at 6:23
What about the case that part of $a_n$ are positive and part of it are negative – Mathematics Sep 10 '12 at 6:54

If $\sum a_n$ converges absolutely, the the answer is affimative. We claim that this is no longer the case for conditional convergence. Note that

$$\frac{x}{1+|x|} = x - x|x| + O(x^3)$$

near the origin. Now consider the series

$$\sum_{n=1}^{\infty} a_n = \frac{2}{\sqrt{1}} - \frac{1}{\sqrt{1}} - \frac{1}{\sqrt{1}} + \frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} + \frac{2}{\sqrt{3}} - \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{3}} + \cdots.$$

This series converges conditionally. Now then we have

$$\frac{a_{3n-2}}{1+|a_{3n-2}|} + \frac{a_{3n-1}}{1+|a_{3n-1}|} + \frac{a_{3n}}{1+|a_{3n}|} = -\frac{2}{n} + O\left( \frac{1}{n^{3/2}}\right).$$

Therefore the sum $\sum \frac{a_n}{1+|a_n|}$ diverges.

Slightly modifying this argument also generates a conditionally convergent series $\sum a_n$ whose corresponding sum $\sum \frac{a_n}{1+|a_n|}$ also converges, thus the answer is inconclusive.

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Surprisingly, it need not be convergent. In fact, if $f$ is a function such that $\sum_n a_n$ converges iff $\sum_n f(a_n)$ converges, then $f$ must be linear in some neighbourhood of $0$. See https://groups.google.com/forum/?hl=en&fromgroups=#!topic/sci.math.research/SzZDXRFyvhk

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Unless I am mistaken, the main information this forum page contains is that This was proved by G. Waldenberg, American Mathematical Monthly 95 (1988) 542-544. Y. Benjamini's solution to Problem E3404, American Mathematical Monthly 99 (1992) 466-467 contains an extension. Let me suggest that you include this in your answer. – Did Sep 10 '12 at 8:03