EXAMPLE $2$ The series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{(-1{)}^{n}7n}{10n-2}$ is alternating, but

$\underset{n}{lim}{b}_n=\underset{n}{lim}\frac{7n}{10n-2}=\underset{n}{lim}\sqrt[\phantom{2}]{\frac{7}{}}=$

so condition $($i$)$ in the Alternating Series Test, that the limit of the terms approaches $0,$ is not sach at the limit of the $n$th term of the series:

$\underset{n}{lim}{a}_n=\underset{n}{lim}\frac{(-1{)}^n{7}_n}{10n-2}$

This limit does not exist, so the series diverges by the Test for Divergence.