Calculus How To

Root Test: Definition, Simple Examples

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Series Convergence Tests >

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:
use the root test on a series

First, calculate the limit:
root test

  • 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:
root test

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.

root test

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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Stephanie Glen. "Root Test: Definition, Simple Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/root-test/
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