$\lim\limits_{n\to\infty}\frac{x^{n+1}/(n+1)^{n+1}}{x^n/n^n}$

$=\lim\limits_{n\to\infty}\frac{x}{n+1}\cdot\Big(\frac{n}{n+1}\Big)^n$

$\lim\limits_{n\to\infty}\frac{xe}{n+1}=?$

$\lim\limits_{n\to\infty}\frac{(n+1)^2x^{n+1}/2^{n+1}}{n^2x^n/2^n}$

$=\lim\limits_{n\to\infty}\frac{x(n+1)^2}{2n^2}=\frac{x}{2}$

For the radius of convergence:

$\Big|\frac{x}{2}\Big|\le1$

To determine the interval of convergence, check if the endpoints ($x = 2$ and $x = -2$) are open or closed. Check each series generated at the endpoints:

$\displaystyle\sum_{n=1}^{\infty}\frac{n^2\cdot2^n}{2^n}\text{ and }\sum_{n=1}^{\infty}\frac{n^2\cdot(-2)^n}{2^n}$

$\lim\limits_{n\to\infty}\frac{x^{3n+3}/(3n+3)!}{x^{3n}/(3n)!}$

$\lim\limits_{n\to\infty}\frac{x^3}{(3n+3)(3n+2)(3n+1)}=?$