Answer
It is not necessary that the series $\Sigma(a_{n} +b_{n})$ will be divergent.
Work Step by Step
Consider
$\Sigma^{\infty}_{n=1} a_{n} = \Sigma^{\infty}_{n=1}1$ and $\Sigma^{\infty}_{n=1}b_{n} = \Sigma^{\infty}_{n=1} -1$
They are both divergent series.
However,
$\Sigma^{\infty}_{n=1}(a_{n} +b_{n}) = \Sigma^{\infty}_{n=1}(1+(-1)) = \Sigma^{\infty}_{n=1}0 =0$ is a convergent series.
It is not necessary that the series $\Sigma(a_{n} +b_{n})$ will be divergent.
