Sum of convergent and non-convergent series, does it converge? And how to prove? Series $a_n$ is convergent and $b_n$ is not-convergent. Will the sum $a_n + b_n$ converge? I think it will not converge, But how do I show it?
I believe I have to use the definition.
$|a_n - A| < \epsilon$
$|b_n - B| >= \epsilon$
Then choose $N > n$ for both, and try to achieve and equation that shows $ | a_n + b_n - A - B | >= \epsilon $

Hints:
By Contradiction, say $\sum ( a_n + b_n) $ converges, and write
$$ - \sum a_n + \sum (a_n + b_n) = \sum b_n $$
What happens now?
