If $\sum a_n$ is a convergent series of real numbers, then is $\sum \dfrac {a_n}{1+|a_n|}$ also convergent? I have tried Abel's, Dirichlet's test; nothing's working.
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Since $f(x)=x/(1+|x|)$ is not linear in any neighborhood of $0$, the answer is negative. – Dec 29 '15 at 06:49
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How could this be false when each term of the second series is less than or equal to the term in the first? – pancini Dec 29 '15 at 06:53
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@ElliotG If $a_n$ is non-negative, then the statement is true but the comparison test doesn't give anything for general series. – levap Dec 29 '15 at 06:56
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@levap thanks; I thought I must be missing something obvious – pancini Dec 29 '15 at 06:57
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10Statement is false, a simple counterexample: $$a_n = \begin{cases} +\frac{1}{\sqrt{k}}, & n = 3k+1 \text{ or } 3k+2\ -\frac{2}{\sqrt{k}}, & n = 3k \end{cases}$$ – achille hui Dec 29 '15 at 07:08
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1@achille hui That's worth writing up as an answer. – zhw. Dec 29 '15 at 07:21
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Achille hui i tried your example but it not work. Can you explain more 3. – CHOUDHARY bhim sen Oct 29 '20 at 07:12