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