The limit laws for series are almost identical to the limit laws for sequences and for functions. If $\displaystyle\sum_{n=1}^\infty a_n = L$ and $\displaystyle\sum_{n=1}^\infty b_n = M$ (i.e., the series converge), then

There are no limit laws for $\displaystyle\sum_{n=1}^\infty\left( a_nb_n\right)$ or $\displaystyle\sum_{n=1}^\infty \left(\frac{a_n}{b_n}\right)$.

Notice that

If $\displaystyle\sum_{n=1}^\infty a_n$ converges, then $\displaystyle\lim_{n \to \infty} a_n = 0$.

Conversely, if $\displaystyle\lim_{n \to \infty} a_n$ is not zero (or does not exist), then $\displaystyle\sum_{n=1}^\infty a_n$ diverges. (This is also called the Divergence test.)