# Chapter 56: The Ratio Test

• Let ﻿$\rho = \lim_{n\rightarrow \infty} \frac{a_n +1}{a_n}$﻿
• ﻿$|\rho| < 1 \Rightarrow \sum_{n=1}^{\infty}a_n$﻿ converges
• ﻿$|\rho| > 1 \Rightarrow \sum_{n=1}^{\infty}a_n$﻿ diverges
• ﻿$|\rho| = 1$﻿ means that the test is inconclusive

#### example #1

• Determine if ﻿$\sum_{n=1}^{\infty}(\frac{1}{2})^a$﻿ converges
• Apply the ratio test #### example #2

• Show that ﻿$\sum_{n=1}^{\infty}\frac{1}{n!}$﻿ converges #### example #3

• Determine whether ﻿$\sum_{n=1}^{\infty}\frac{(2n)!}{n!n!}$﻿ converges #### Example #4

• Discuss whether ﻿$\sum_{n=1}^{\infty}\frac{n^2}{(2n)!}$﻿ diverges
• ﻿$\rho = \lim_{n \rightarrow \infty} \frac{(n+1)^2)}{(2n+2)!} \cdot \frac{(2n)!}{n^2}$﻿
• ﻿$= \lim_{n \rightarrow \infty} \frac{(n+1)^2}{(2n+2)(2n+1)n^2} = 0$﻿

### The Root Test

• Let ﻿$C = \lim_{n\rightarrow \infty} \sqrt[n]{19_n1}$﻿
• If ﻿$C < 1$﻿ then ﻿$\sum_{n=1}^{\infty}a_n$﻿ converges
• If ﻿$C > 1$﻿ then ﻿$\sum_{n=1}^{\infty}a_n$﻿ diverges
• If ﻿$C = 1$﻿ then the test is inconclusive

#### Example #5

• Determine whether ﻿$\sum_{n=1}^{\infty}\frac{n^2}{2^n}$﻿ converges
• Apply the root test • Calculate ﻿$\lim_{n\rightarrow \infty} \sqrt[n]{n^2}$﻿ • Calculate ﻿$\lim_{n\rightarrow \infty} \frac{2}{n}ln(n)$﻿ #### example #6

• Determine whether ﻿$\sum_{n=1}^{\infty}(\frac{1}{n+1})^1$﻿ converges
• Apply the root test • A series ﻿$\sum_{n=1}^{\infty}a_n$﻿ converges absolutely if ﻿$\sum_{n=1}^{\infty} |a_n|$﻿ converges

#### example #7

• ﻿$\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$﻿ converges but does not converge absolutely #### example #8

• Check that ﻿$\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$﻿ converges and converges absolutely 