# The Root Test For Series Convergence

## Series Convergence Tests

A **Series** is a summation of infinitely many quantities given in a certain order. Consider the following expression:

$$\sum_{i=1}^{\infty} a_i $$

Any expression of this form is called a series. It's also represented as:

$$a_{1}+a_{2}+\cdots+a_{n}+\cdots $$

In connection to a series, the definition of a certain term is called **sequence of partial sums**.

Given a series of the form {eq}\sum_{i=1}^{\infty} a_i {/eq} a term in the sequence of partial sums {eq}S_{n} {/eq} is defined as the sum of first {eq}n {/eq} terms of the series. That is:

$$S_{n}\sum_{i=1}^{n} a_i $$

A series is said to converge to a certain real number, {eq}s {/eq}, if its sequence of partial sums converges to the real number {eq}s {/eq}. Such a series is called **convergent**.

When the sequence of partial sums diverges, the series is called **divergent**.

There are many tests to check whether a series converges or diverges. The most commonly used series convergence tests are:

- The Root Test
- The Ratio Test
- The Comparison Test
- The Integral Test

Knowing whether a series converges or diverges gives a definite way of working with it. When it's known that a series diverges, it is said that the series grows without bound and hence can be used in accordance. Different tests work in different manners on each series. To elaborate, some cases exist where one test fails to accomplish whether a certain series converges or diverges while using another gives us a definite answer.

## The Root Test

If you know that a series converges, then you can work further on it. But if it doesn't converge, then you can stop working on the series because you won't find an end to it. So how can you tell? Well, there's a test you can run.

The **root test** is a simple test that tests for absolute convergence of a series, meaning the series definitely converges to some value. This test doesn't tell you what the series converges to, just that your series converges.

The formal statement for the root test is:

For a series made up of terms *a**n*, define the limit as follows in this equation:

We then keep the following in mind:

- If
*L*< 1, then the series absolutely converges. - If
*L*> 1, then the series diverges. - If
*L*= 1, then the series is either divergent or convergent.

That last statement basically means that if you get 1 for your *L* then your answer is unknown. The root test can't tell whether your series converges or diverges.

Now, let's take a look at using the root test for a converging series, a diverging series, and an unknown or indeterminate series.

## The Root Test for Series Convergence

The root test for series convergence states that:

Let {eq}\sum_{i=1}^{\infty} a_{i} {/eq} be a series of real numbers. Define {eq}L {/eq} as the limit:

$$L=\lim_{n\to \infty}\sqrt[n]{|a_n|}=\lim_{n\to \infty}(|a_n|)^{\frac{1}{n}} $$

Then:

- {eq}L<1 {/eq} implies the series converges.

- {eq}L>1 {/eq} implies the series diverges.

- {eq}L=1 {/eq} implies the test fails.

So if the value of {eq}L {/eq} is less than or greater than one, then there is a definite answer to the question of convergence, but when it's equal to one the test fails to inform about the convergence of the series and thus needs some other method of determination.

Applying the root test to the sequence part of a power series gives us the interval of convergence of a power series.

The root test series is considered to be one of the strongest among all series convergence tests.

The proof for the root test is as follows:

Let us consider each case:

- {eq}L<1 {/eq}:

{eq}L\geq 0 {/eq} always holds as it is the limit of the sequence of positive numbers.

Since {eq}L<1 {/eq} choose {eq}r > 0 {/eq} such that {eq}0<L+r<1 {/eq}( such an {eq}r {/eq} always exists, for example consider {eq}r=\frac{1-L}{2} {/eq}.

Since the sequence {eq}|a_n|^{\frac{1}{n}} {/eq} converges to {eq}L {/eq} we can find {eq}N \in \mathbb{N} {/eq} such that:$$\forall n \geq N \\ |a_{n}|^{\frac{1}{n}}<L+r $$

Therefore we see that:

$$\forall n \geq N \\ |a_{n}|<(L+r)^n $$.

Hence we see that

$$\forall n \geq N \\ a_{n}<(L+r)^n $$

This, in turn, implies that

$$\sum_{n=N}^{\infty}a_{n}<\sum_{n=N}^{\infty}(L+r)^n $$

The series on the right-hand side converges as it's the geometric series with its common ratio of less than one.

Consequently the series {eq}\sum_{n=N}^{\infty}a_{n} {/eq} converges by comparison test (given two series, if the greater one converges then so does the lesser).

Now going back to the series we see that:

$$\sum_{n=1}^{\infty}a_{n}= \sum_{n=1}^{N-1}a_{n}+\sum_{n=N}^{\infty}a_{n} $$

The first term in the sum on the right-hand side is the sum of finitely many real numbers and the second is a convergent series, Hence our series {eq}\sum_{n=1}^{\infty}a_{n} {/eq} converges.

- {eq}L > 1 {/eq}

Given

$$|a_{n}|^\frac{1}{n} \to L>1, n \to \infty $$

Corresponding to this we see that there exists some neighborhood of zero which contains only finitely many terms of the sequence {eq}a_{n} {/eq}. Therefore {eq}a_{n} {/eq} can never converge to zero. If the series converges, the limit of its corresponding sequence terms should be zero. Therefore our series does not converge when {eq}L > 1 {/eq}.

The third case is discussed in the examples in the final section.

## How to Tell if a Series Converges or Diverges Using the Root Test

We will see how to tell whether a series converges or diverges using the following example:

Consider the series:

$$\sum_{n=1}^{\infty}\left(\frac{n^2+3n}{4n^2+5n}\right)^n $$

Applying the root test:

$$\begin{aligned} L&=\lim_{n\to\infty}\sqrt[n]{\left|\frac{n^2+3n}{4n^2+5n}\right|^n}\\&=\lim_{n\to\infty}{\left|\frac{n^2+3n}{4n^2+5n}\right|^n}^{\frac{1}{n}}\\&=\lim_{n\to\infty}{\left|\frac{n^2+3n}{4n^2+5n}\right|}\\&=\frac{1}{4}<1 \end{aligned} $$

Hence here {eq}L<1 {/eq} therefore the series converges.

Keep in mind how the powers are canceled off while applying limits. This is a very important aspect of the test, it works best when {eq}n^{th} {/eq} powers are present in the series term.

In the next section, we will see different cases where the root test can be applied. There are cases where the root test for convergence fails. In such examples, we will have to make use of other series convergence tests.

## Examples of the Root Test

Consider the following root test examples.

### Example 1

Consider the series:

$$\sum_{n=1}^{\infty}\left(\frac{3}{n+1}\right)^n $$

Applying the root test:

$$\begin{aligned} L&=\lim_{n\to\infty}\sqrt[n]{\left|\frac{3}{n+1}\right|^n}\\&=\lim_{n\to\infty}{\left|\frac{3}{n+1}\right|^n}^{\frac{1}{n}}\\&=\lim_{n\to\infty}{\left|\frac{3}{n+1}\right|}\\&=0<1 \end{aligned} $$

Hence here {eq}L<1 {/eq} therefore the series converges.

### Example 2

Consider the series:

$$\sum_{n=1}^{\infty}\left(\frac{n}{\ln(n)}\right)^n $$

Applying the root test we see that:

$$\begin{aligned} L&=\lim_{n\to\infty}\sqrt[n]{\left|\frac{n}{\ln(n)}\right|^n}\\&=\lim_{n\to\infty}{\left|\frac{n}{\ln(n)}\right|^n}^{\frac{1}{n}}\\&=\lim_{n\to\infty}{\left|\frac{n}{\ln(n)}\right|}\\&=\infty>1 \end{aligned} $$

Hence here {eq}L>1 {/eq} therefore the series diverges.

### Example 3

Consider the series:

$$\sum_{n=1}^{\infty}(1)^n $$

It's easily seen that the series is divergent as the sequence of partial sums is the set of all natural numbers which does not converge (as it has no bound).

Applying root test we see that:

$$\begin{aligned} L&=\lim_{n\to\infty}\sqrt[n]{(1)^n}\\&=\lim_{n\to\infty}{(1)^n}^{\frac{1}{n}}\\&=\lim_{n\to\infty}{1}\\&=1 \end{aligned} $$

Hence here {eq}L=1 {/eq} therefore the root test fails.

But the series is divergent.

### Example 4

Consider the series:

$$\sum_{n=1}^{\infty}\frac{1}{n^2} $$

This is a convergent series. But on applying the root test:

$$\begin{aligned} L&=\lim_{n\to\infty}\sqrt[n]\frac{1}{n^2}\\&=1 \end{aligned} $$

Hence here too the root test fails but the series is convergent.

In the last two examples, we see that the root test for convergence fails. In such cases, we have to make use of tests other than the root test series.

## Lesson Summary

A **Series** is a summation of infinitely many quantities given in a certain order. Consider the following expression:

$$\sum_{i=1}^{\infty} a_i $$

Any expression of this form is called a series.

In connection to a series, we define a certain term called **sequence of partial sums**.

Given a series of the form {eq}\sum_{i=1}^{\infty} a_i {/eq} a term in the sequence of partial sums {eq}S_{n} {/eq} is defined as the sum of first {eq}n {/eq} terms of the series. That is:

$$S_{n}\sum_{i=1}^{n} a_i $$

A series is said to converge to a certain real number, {eq}s {/eq}, if its sequence of partial sums converges to the real number {eq}s {/eq}. Such a series is called **convergent**.

When the sequence of partial sums diverges we call the series **divergent**.

Let {eq}\sum_{i=1}^{\infty} a_{i} {/eq} be a series of real numbers. Define {eq}L {/eq} as the limit:

$$L=\lim_{n\to \infty}\sqrt[n]{|a_n|}=\lim_{n\to \infty}(|a_n|)^{\frac{1}{n}} $$

Then:

- {eq}L<1 {/eq} implies the series converges.

- {eq}L>1 {/eq} implies the series diverges.

- {eq}L=1 {/eq} implies the test fails.

So if the value of {eq}L {/eq} is less than or greater than one then we have a definite answer to the question of convergence but when it's equal to one we see that the test fails to inform us about the convergence of the series and hence needs some other method of determination.

Applying the **root test** to the sequence part of a power series gives us the interval of convergence of a power series. Here the L value calculated gives the radius of the interval of convergence.

## Converging Series

First, let's look at a converging series. Here's the problem:

Use the root test to determine whether this series converges or diverges.

To use the root test, you'll follow the statement for the root test and take the limit of the absolute value of the terms in the series taken to the 1 / *n* power like this series of equations appearing here:

The *L* is equal to 4 / 5, which is less than 1. Therefore, according to the root test, this series absolutely converges. Notice how the power of 1 / *n* was canceled since the series you are working with has an *n* power already. After that, you take the limit by plugging in infinity into your *n* variable and then canceling as much as you can, leaving you with 4 / 5.

## Divergent Series

Now, let's look at a diverging series. Does this series diverge or converge? Use the root test.

Then you use the root test and things should look like this:

Do the same steps that you did before. This time, though, you get a 4 as your limit. Since this 4 is greater than 1, then this series diverges, according to the root test.

## One More Example

This last example is of a series that the root test won't work on. If you apply the root test, you'll get an unknown or indeterminate answer.

Again, you need to use the root test to determine whether this series diverges or converges.

Using the root test, you get this:

The limit here equals 1, so this series is unknown or indeterminate. Your series can diverge or converge; the root test can't tell. You'll have to use another method to figure it out.

## Lesson Summary

All right, let's review. The **root test** is a simple test that tests for absolute convergence of a series. For a series made up of terms *a**n*, define the limit as follows:

We then learned that we have to keep the following in mind:

- If
*L*< 1, then the series absolutely converges. - If
*L*> 1, then the series diverges. - If
*L*= 1, then the series is either divergent or convergent.

You should now have an easier time solving a series convergence now that you know how to use the root test and understand what the results mean.

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Create your account

## The Root Test

If you know that a series converges, then you can work further on it. But if it doesn't converge, then you can stop working on the series because you won't find an end to it. So how can you tell? Well, there's a test you can run.

The **root test** is a simple test that tests for absolute convergence of a series, meaning the series definitely converges to some value. This test doesn't tell you what the series converges to, just that your series converges.

The formal statement for the root test is:

For a series made up of terms *a**n*, define the limit as follows in this equation:

We then keep the following in mind:

- If
*L*< 1, then the series absolutely converges. - If
*L*> 1, then the series diverges. - If
*L*= 1, then the series is either divergent or convergent.

That last statement basically means that if you get 1 for your *L* then your answer is unknown. The root test can't tell whether your series converges or diverges.

Now, let's take a look at using the root test for a converging series, a diverging series, and an unknown or indeterminate series.

## Converging Series

First, let's look at a converging series. Here's the problem:

Use the root test to determine whether this series converges or diverges.

To use the root test, you'll follow the statement for the root test and take the limit of the absolute value of the terms in the series taken to the 1 / *n* power like this series of equations appearing here:

The *L* is equal to 4 / 5, which is less than 1. Therefore, according to the root test, this series absolutely converges. Notice how the power of 1 / *n* was canceled since the series you are working with has an *n* power already. After that, you take the limit by plugging in infinity into your *n* variable and then canceling as much as you can, leaving you with 4 / 5.

## Divergent Series

Now, let's look at a diverging series. Does this series diverge or converge? Use the root test.

Then you use the root test and things should look like this:

Do the same steps that you did before. This time, though, you get a 4 as your limit. Since this 4 is greater than 1, then this series diverges, according to the root test.

## One More Example

This last example is of a series that the root test won't work on. If you apply the root test, you'll get an unknown or indeterminate answer.

Again, you need to use the root test to determine whether this series diverges or converges.

Using the root test, you get this:

The limit here equals 1, so this series is unknown or indeterminate. Your series can diverge or converge; the root test can't tell. You'll have to use another method to figure it out.

## Lesson Summary

All right, let's review. The **root test** is a simple test that tests for absolute convergence of a series. For a series made up of terms *a**n*, define the limit as follows:

We then learned that we have to keep the following in mind:

- If
*L*< 1, then the series absolutely converges. - If
*L*> 1, then the series diverges. - If
*L*= 1, then the series is either divergent or convergent.

You should now have an easier time solving a series convergence now that you know how to use the root test and understand what the results mean.

To unlock this lesson you must be a Study.com Member.

Create your account

- Activities
- FAQs

## Root Test for Convergence: True or False Activity

This activity will help assess your knowledge of the root test and how it is used as a test for convergence.

### Directions

Print or copy this page on blank paper. Then, write TRUE if the given statement is correct and FALSE if otherwise. Neatly write your answers on the appropriate blank space provided. Lastly, use the picture below as your reference.

__________ 1. In the root test, the convergence/divergence of the series is decided by the value of the limit *L*.

__________ 2. An *L* of positive infinity indicates that the series absolutely converges.

__________ 3. Taking the root test of series *A* will result in *L* equal to infinity.

__________ 4. Series *A *and *D* diverge.

__________ 5. The limit *L* can have positive and negative values.

__________ 6. A series diverges when its value gets closer and closer to a given limit or number.

__________ 7. Series *D* is either divergent or convergent.

__________ 8. Series *B* and *C* are both lower than one and would absolutely converge.

### Answer Key

1. TRUE

2. FALSE, because the correct statement is: An *L* of positive infinity indicates that the series diverges.

3. TRUE

4. FALSE, because the correct statement is: Only series *A* diverges.

5. FALSE, because the correct statement is: The limit *L* can only have positive values.

6. FALSE, because the correct statement is: A series converges when its value gets closer and closer to a given limit or number.

7. TRUE

8. TRUE

### Solution

#### Does the root test show absolute convergence?

Yes, the root test shows that the series is absolutely convergent and also converges conditionally. We use the modulus of the series terms, therefore, it gives conclusive proof that series converges absolutely.

#### Can you use the root test to find the interval of convergence?

Yes, in the case of a power series, applying the root test to the sequence in the power series gives the interval of convergence. Here the L value determines the radius of the interval of convergence.

#### What is the difference between the ratio test and the root test?

The ratio test uses the ratio between successive terms of the sequence to determine the convergence of the series. The root test employs the root of the nth term to do the same.

#### When can you use the root test?

The root test is most effective when used in series with nth power terms. It is of little help if the series contains factorials.

#### What is the root test for convergence?

The root test states that given a series with the limit of the sequence of nth roots of the nth term of the sequence that forms the series is less than one, then the series converges. If the limit is greater than one then the series diverges, and if the limit is equal to one then the test is inconclusive.

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