Answer
See below.
Work Step by Step
Theorem 8 deals with these types of properties for $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t}$ series:
If $\displaystyle \sum_{n=1}^{\infty}a_{n}=A$ and $\displaystyle \sum_{n=1}^{\infty}b_{n}=B$ are convergent series, then
1. Sum Rule:
$\displaystyle \sum_{n=1}^{\infty}(a_{n}+b_{n})=\sum_{n=1}^{\infty}a_{n}+\sum_{n=1}^{\infty}b_{n}=A+B.$
2. Difference Rule:
$\displaystyle \sum_{n=1}^{\infty}(a_{n}-b_{n})=\sum_{n=1}^{\infty}a_{n}-\sum_{n=1}^{\infty}b_{n}=A-B.$
3. Constant Multiple Rule:
$\displaystyle \sum_{n=1}^{\infty}ka_{n}=k\sum_{n=1}^{\infty}a_{n}=kA$ for any number $k.$
And, if the series in question are $\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t}$, we have the Corollary to theorem 8:
1. Every nonzero constant multiple of a divergent series diverges.
2. If $\displaystyle \sum_{=1}^{\infty}a_{n}$ converges and $\displaystyle \sum_{n=1}^{\infty}b_{n}$ diverges,
then $\displaystyle \sum_{n=1}^{\infty}(a_{n}+b_{n})$ and $\displaystyle \sum_{n=1}^{\infty}(a_{n}-b_{n})$ both diverge.
