### Home > APCALC > Chapter 12 > Lesson 12.1.4 > Problem12-47

12-47.

Determine the interval of convergence for each of the following series.

1. $\displaystyle \sum _ { n = 0 } ^ { \infty } \frac { 1 } { ( 2 n ) ! } x ^ { 2 n }$

Once the Ratio Test is first applied, your result should be:

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

1. $\displaystyle\sum _ { n = 2 } ^ { \infty } n ^ { 2 } ( x + 1 ) ^ { n }$

Once the Ratio Test is applied and simplified, your result should be:

$|x+1|\le 1$

Once you solve for $x$, be sure to check the endpoints.

1. $\displaystyle \sum _ { n = 1 } ^ { \infty } ( \frac { n } { 2 n - 5 } ) ^ { n } x ^ { n }$

$\lim \limits_{n\to\infty}\Bigg|\frac{\Big(\frac{n+1}{2n-3}\Big)^{n+1}x^{n+1}}{{\Big(\frac{n}{2n-5}\Big)^nx^n}}\Bigg|$

$\lim \limits_{n\to\infty}\bigg|\Big(\frac{(2n-5)(n+1)}{(2n-3)n}\Big)^n\Big(\frac{n+1}{2n-3}\Big)x\bigg|$

$\bigg|\frac{1}{2}x\bigg|\le1$

Once you solve for $x$, be sure to check the endpoints.