$(1)$ Consider the power series

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{(-1{)}^n(x-4{)}^n}{3^n\sqrt[\phantom{2}]{n}}$

$($a$)(2$ points$)$ Use the root or ratio test to determine the largest open interval on which this series converges absolutely$.$ You should obtain a finite interval, not $(-,).$

$($b$)(2$ points$)$ Test each of the endpoints of your interval in part $($a$).$ Does the series converge at either one? If yes, is the convergence absolute or conditional?