  ### Home > APCALC > Chapter 10 > Lesson 10.3.2 > Problem10-142

10-142.

Determine the radius and interval of convergence for each of the following power series.

1. $\displaystyle \sum _ { n = 1 } ^ { \infty } ( \frac { x } { n } ) ^ { n }$

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

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

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

1. $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } x ^ { n } } { 2 ^ { n } }$

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

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

$\Big|\frac{x}{2}\Big|\le1$
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:
$\displaystyle\sum_{n=1}^{\infty}\frac{n^2\cdot2^n}{2^n}\text{ and }\sum_{n=1}^{\infty}\frac{n^2\cdot(-2)^n}{2^n}$
1. $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { x ^ { 3 n } } { ( 3 n ) ! }$
$\lim\limits_{n\to\infty}\frac{x^{3n+3}/(3n+3)!}{x^{3n}/(3n)!}$
$\lim\limits_{n\to\infty}\frac{x^3}{(3n+3)(3n+2)(3n+1)}=?$