Does there exist a series $\sum_{n=1}^{\infty} b_n$ such that $\sum_{i=1}^{\infty} b_{n_i}$ converges for any $n_1 < n_2 < \ldots$ but $\sum_{n=1}^{\infty} |b_n|$ diverges?
Certainly the example would have to be conditionally convergent, but such standard examples as the alternating harmonic series don't seem to work (at least from what I've managed to show).
Any ideas?
No. Define $$b_{n_i}=b_n^+=\max\{b_i,0\}.$$ By hypothesis, $\sum_{i=0}^\infty b_{n_i}$ converges. Similarly, we can define $b_n^-$ and see that the sum over $b_n^-$ is convergent by hypothesis.
This means that $\sum_{n=0}^\infty b_n$ is absolutely convergent since both its positive parts and negative parts converge.