Answer
$R=\frac{5}{2}$ ; interval of convergence is $[-2,3)$
Work Step by Step
Let $a_{n}=\frac{(2x-1)^{n}}{5^{n}\sqrt n}$, then
$\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\frac{\frac{(2x-1)^{n+1}}{5^{n+1}\sqrt n}}{\frac{(2x-1)^{n}}{5^{n}\sqrt n}}|$
$=\frac{|2x-1|}{5}$
$=\frac{|2x-1|}{5}\lt 1$
$-2\lt x\lt 3$
Hence, $R=\frac{5}{2}$ ; interval of convergence is $[-2,3)$
