If two positive terms series $\sum_{n=1}^{\infty} a_n, \sum_{n=1}^{\infty} b_n$ are divergent, $\sum_{n=1}^{\infty} (a_n+b_n)$ is also divergent.
I thought is was obvious, but I saw a counterexample of this problem, that is, $a_n = n, b_n = -n.$ However, this is a little bit strange, because $b_n$ is NOT positive terms series.
What's wrong with my thoughts?
If both are positive then yes, your thoughts are correct, for example by direct comparison. That example isn't relevant because, as you said, $b_n$ is not a positive sequence.