[Series] Ratio Test

2021-10-13 19:19 作者:AoiSTZ23 0人读过 | 我要投稿

By: Tao Steven Zheng (郑涛)

【Problem】

Part 1: Consider the following infinite series. Determine whether each infinite series converges or diverges.

（1） $%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cfrac%7Bn%7D%7B2%5En%7D%20$

（2） $%20%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cfrac%7B2%5En%7D%7Bn%7D%20$

Part 2: Consider the following infinite series. Determine the radius of convergence and the interval of convergence.

$%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cfrac%7B4%5En%7D%7Bn%7D(x-3)%5En%20$

【Solution】

Part 1
apply the ratio test to each series.
（1）
$%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cleft%7C%20%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%20%5Cright%7C%20$

$%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cleft%7C%20%5Cfrac%7B%5Cfrac%7Bn%2B1%7D%7B2%5E%7Bn%2B1%7D%7D%7D%7B%5Cfrac%7Bn%7D%7B2%5En%7D%7D%5Cright%7C$

$%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cleft%7C%20%5Cfrac%7Bn%2B1%7D%7B2n%7D%20%5Cright%7C%20%3D%20%5Cfrac%7B1%7D%7B2%7D$

Since this limit is less than 1, the series converges.

（2）
$%20%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cleft%7C%20%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%20%5Cright%7C%20$

$%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cleft%7C%20%5Cfrac%7B%5Cfrac%7B2%5E%7Bn%2B1%7D%7D%7Bn%2B1%7D%7D%7B%5Cfrac%7B2%5En%7D%7Bn%7D%7D%20%5Cright%7C%20$

$%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cleft%7C%20%5Cfrac%7B2n%7D%7Bn%2B1%7D%20%5Cright%7C%20%3D%202%20$

Since this limit is greater than 1, the series diverges.

Part 2
In order for the series to converge, the limit must be less than 1.

$%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cleft%7C%20%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%20%5Cright%7C%20%3C%201%20$

For the series $%20%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cfrac%7B4%5En%7D%7Bn%7D(x-3)%5En$ ,

$%20%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cleft%7C%20%5Cfrac%7B4%5E%7Bn%2B1%7D%7D%7Bn%2B1%7D%20%7B(x-3)%7D%5E%7Bn%2B1%7D%20%5Ccdot%20%5Cfrac%7Bn%7D%7B4%5En%20%7B(x-3)%7D%5E%7Bn%7D%7D%20%5Cright%7C%20%3C%201$

$%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cleft%7C%20%5Cfrac%7B4n%7D%7Bn%2B1%7D(x-3)%20%5Cright%7C%20%3C%201$

$%20%5Clim_%7Bn%5Cto%5Cinfty%7D%20%5Cleft%7C%20%5Cfrac%7B4n%7D%7Bn%2B1%7D%5Cright%7C%20%7Cx-3%7C%20%3C%201$

$%204%7Cx-3%7C%20%3C%201%20$

$%20%7Cx-3%7C%20%3C%20%5Cfrac%7B1%7D%7B4%7D$

At this stage, we find that the radius of convergence is $%20R%20%3D%20%5Cfrac%7B1%7D%7B4%7D$ . To solve for the interval of convergence, solve for $x$ .

$%5Cfrac%7B-1%7D%7B4%7D%20%3C%20x-3%20%3C%20%5Cfrac%7B1%7D%7B4%7D$

$%5Cfrac%7B11%7D%7B4%7D%20%3C%20x%20%3C%20%5Cfrac%7B13%7D%7B4%7D%20$

This is as far as the ratio test allows us to conclude. To determine if each endpoint converges or not, we must apply other tests.

When $x%3D%5Cfrac%7B11%7D%7B4%7D$ , the series becomes

$%20%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cfrac%7B4%5En%7D%7Bn%7D%7B%5Cleft(%5Cfrac%7B-1%7D%7B4%7D%5Cright)%7D%5E%7Bn%7D$

$%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cfrac%7B(-1)%5En%7D%7Bn%7D%20$

By the alternating series test, this series converges.

When $x%20%3D%20%5Cfrac%7B13%7D%7B4%7D$ , the series becomes

$%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cfrac%7B4%5En%7D%7Bn%7D%7B%5Cleft(%5Cfrac%7B1%7D%7B4%7D%5Cright)%7D%5E%7Bn%7D$