Test the series for convergence or divergence using the Alternating Series Test.

${\displaystyle \underset{n=1}{\overset{}{}}}(-1{)}^{n-1}{b}_n=\frac{1}{\sqrt[\phantom{2}]{4}}-\frac{1}{\sqrt[\phantom{2}]{5}}\frac{1}{\sqrt[\phantom{2}]{6}}-\frac{1}{\sqrt[\phantom{2}]{7}}\frac{1}{\sqrt[\phantom{2}]{8}}-$dots

Identify $b_n.$

Evaluate the following limit$.$

$\underset{n}{lim}{b}_n$

Since $\underset{n}{lim}{b}_{n}0$ and $b_{n1}{b}_n$ for all $n,.$