The root test tells us whether or not a series converges absolutely.

As its name suggests, it involves taking a root, and so it is most useful for exponential series and in situations where taking a root is quick and simple.

## Determining Absolute Convergence Using the Root Test

To use the root test on a series:

First, calculate the limit:

- If that limit is less than one, the series converges absolutely.
- If it is greater than one, it diverges.
- And if the limit is just one, we don’t know. It might converge conditionally, it
*might*diverge.

There are other tests which can be used to determine convergence when the root test gives us an indeterminate answer.

## Examples of the Root Test

To see how this works, suppose you are given the series:

We need to take the limit, as n goes to infinity, of the nth root of that. *Taking the nth root* divides any exponent by 1/n. In the above formula, there are two exponents (*n* in the numerator and *1 + 2n* in the denominator); each of these needs to be divided by 1/n.

Using that information, you should get:

Since infinity is greater than 1, this series is divergent.

The following series is a little more interesting:

Here, again, we divide the exponents by 1/n.

Since 12/10 is also greater than one, this series also diverges.

## References

Belk, Jim. Math 142 Course Notes. Root Test. Bard College Math Department. Published Online Fall 2009. Retrieved from http://faculty.bard.edu/belk/math142af09/RootTest.pdf on August 26, 2019.

Oregon State Calculus Quest. Series, Convergence, and Series Tests. Retrieved from https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/SandS/SeriesTests/root.html on August 26, 2019

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