Chapter 11 - Infinite Sequences and Series - 11.8 Exercises - Page 770: 31

$R=k^{k}$

Let $a_{n}=\frac{(n!)^{k}x^{n}}{(kn)!}$, then $\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\dfrac{\frac{((n+1)!)^{k}x^{n+1}}{(k(n+1))!}}{\frac{(n!)^{k}x^{n}}{(kn)!}}|$ $=|x|(\frac{1}{k})^{k}$ $=|x|(\frac{1}{k})^{k}\lt 1$ Hence, $R=k^{k}$