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If $\sum a_n$ is convergent then is it true that $\sum a_n/(1+|a_n|)$ convergent? If $a_n$ is positive term then we can easily prove this series is convergent by comparison test but how to proceed for general series?

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    $\begingroup$ How can we prove this easily if $a_n$ is positive? $\endgroup$
    – RhythmInk
    Oct 23, 2018 at 3:50
  • $\begingroup$ Terms of series are less than $a_n$. $\endgroup$ Oct 23, 2018 at 3:52

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No. A counterexample can be built following the ideas of this answer.

For positive integer $n$, let $a_{3n-2}=a_{3n-1}=\dfrac{1}{2\sqrt{n}}$ and $a_{3n}=-\dfrac{1}{\sqrt{n}}$. Then $\displaystyle\sum_{n=1}^{\infty}a_n=0$, but $$\frac{a_{3n-2}}{1+|a_{3n-2}|}+\frac{a_{3n-1}}{1+|a_{3n-1}|}+\frac{a_{3n}}{1+|a_{3n}|}=\frac{1}{(1+\sqrt{n})(1+2\sqrt{n})},$$ thus $\lim_{N\to\infty}\sum_{n=1}^{3N}a_n/(1+|a_n|)=\infty$ and this series diverges.

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