# Root test

### Root test

In this section, we will look at a very useful limit question that will be used frequently when doing the root test. We will then learn and apply the root test to determine the convergence and divergence of series. Root test requires you to calculate the value of R using the formula below. If R is greater than 1, then the series is divergent. If R is less than 1, then the series is convergent. If R is equal to 1, then the test fails and you would have to use another test to show the convergence or divergence of the series. You may notice that this looks very similar to the ratio test. Also note that if the root test fails, then the ratio test will also fail. Thus, make sure to not waste time doing the ratio test if the root test fails.

#### Lessons

Note *Let $\sum a_n$ be a positive series. Then we say that

$R=$ $\lim$n →$\infty$ $\mid a_n\mid^{\frac{1}{n}}$

Where:
1. If $R$ < $1$, then the series is convergent (or absolutely convergent)
2. If $R$ > $1$, then the series is divergent
3. If $R=1$, then the series could either be divergent, or convergent

Basically if $R=1$, then the root test fails and would require a different test to determine the convergence or divergence of the series.

Note that if the root test gives $R=1$, then so will the ratio test.
• Introduction
Root Test Overview

• 1.
Useful Limit Question Used for Root test
Show that $\lim$n →$\infty$ $n^{\frac{1}{n}}=1$. This fact is useful when doing the root test for infinite series.

• 2.
Convergence & Divergence of Root test
Use the Root test to determine if the series converges or diverges. If the root test does not determine the convergence or divergence of the series, then resort to another test.
a)
$\sum_{n=1}^{\infty}\frac{(-3)^n}{2n}$

b)
$\sum_{n=0}^{\infty}\frac{(n)^{2n+1}}{\pi^{1-2n}}$

c)
$\sum_{n=1}^{\infty}[\frac{n^2-2n^3}{5+2n^3} ]^{3n}$

d)
$\sum_{n=1}^{\infty}\frac{n^n}{3^n}$