# Is the sum of positive divergent series always divergent?

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?

• Burning books is generally not a good advice, but if yours has more such "counterexamples"... – Przemysław Scherwentke Dec 25 '14 at 5:13
• how is $\sum b_n$ a positive term series? – Adam Hughes Dec 25 '14 at 5:18
• what do you think is wrong with your thoughts? – Irrational Person Dec 25 '14 at 5:23

## 1 Answer

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.