TITLE: if $\sum a_{n}$ is a convergent series, what about $\sum \frac{a_{n}}{1+|a_{n}|}$?
QUESTION [7 upvotes]: Suppose that $\sum a_{n}$ is convergent series of real numbers. Either prove that $\sum b_{n}$ converges, or give a counter-example, when we define $b_{n}$ by $\frac{a_{n}}{1+|a_{n}|}$.

REPLY [12 votes]: For series that have some negative terms, this is not necessarily true. We give a family of counterexamples.
Consider the series 
$$a_1+a_1-2a_1+a_2+a_2-2a_2+a_3+a_3-2a_3+\cdots,$$
where the $a_i$ are a sequence of positive terms slowly decaying to $0$. The convergence of the above series is clear. 
The modified series need not converge.
For the modified series looks like 
$$\frac{a_1}{1+a_1}+\frac{a_1}{1+a_1}-\frac{2a_1}{1+2a_1}+\frac{a_2}{1+a_2}+\frac{a_2}{1+a_2}-\frac{2a_2}{1+2a_2}+\cdots.$$
If we add terms by threes, we get
$$\sum_{i=1}^\infty \frac{2a_i^2}{(1+a_i)(1+2a_i)}.$$
Choose for instance $a_i=\dfrac{1}{i^{1/8}}$, and we get divergence.