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 $

It is not converge. because $$b_n=(b_n+a_n)-a_n,$$ so if $\sum b_n+a_n $ converges then $b_n$ so is. 
