I believe my proposition is correct, as all of the pairs of series I've tried obey the claim. The strategy for proving this is straightforward. I assume $\sum\limits_{n=0}^{\infty} a_n$ absolutely converges and $\sum\limits_{n=0}^{\infty} b_n$ conditionally converges.
We know $\sum\limits_{n=0}^{\infty} (a_n + b_n)$ converges as both series are assumed to be convergent. The strategy is to show $\sum\limits_{n=0}^{\infty} |a_n + b_n|$ diverges. It's clear the fact $\sum\limits_{n=0}^{\infty} |b_n|$ diverges will be used somewhere, and perhaps we could prove a contradiction by assuming $\sum\limits_{n=0}^{\infty} |a_n + b_n|$ converges, we would derive the conclusion that $\sum\limits_{n=0}^{\infty} |b_n|$ converges.
I've been toying with the triangle inequality for a while but I haven't been able to get anywhere. There's also the possibility the claim is false.
Best Answer
By contradiction, if $\sum |a_n+b_n|$ converges, since $|b_n|-|a_n|\leq |a_n+b_n|$
we have that $\sum |b_n|\leq \sum |a_n| + \sum |a_n+b_n|$
By comparison test, $\sum |b_n|$ converges.
Contradiction.