Sum of two divergence series is always divergence series?

Question

In my opinion, I still believe that sum of two divergence series can be convergence series.

But here is my proof:

First, we have an obvious result: sum of two convergence series is convergence series.

Take opposite of this statement is: if exist one series is (non-convergence) so cannot be (non-convergence)

$==>$ if exist one series is divergence, so sum always be divergence.!!! (because non-convergence series is divergence series and vice verse)

If my proof is wrong, please tell me where and give me an example that sum of two divergence series can be convergence series please.

Thanks :)

Answer 1

If $a_n+b_n = c_n$ and $\sum_n c_n$ is a convergent series, then $b_n=c_n-a_n$, so $\sum_n b_n$ differs from $\sum_n (-a_n)$ by the convergent series $\sum_n c_n$. In particular, $\sum_n b_n$ is convergent if and only if $\sum_n a_n$ is.

If you start with a particular divergent series $\sum_n a_n$, every series you can add termwise to $\sum_n a_n$ to obtain a convergent series has the form $\sum_n(-a_n+c_n)$, where $\sum_n c_n$ is whatever convergent series you like.