In our experience, the Root Test is the least used series test to test for convergence or divergence (which is why it appears last in the infinite series table). The reason is that it is used only in very specific cases, whereas the other tests can be used for a broader range of problems. However, for those specific cases, it can be much easier to use than other techniques to determine convergence or divergence.
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Root Test 

For the series \( \sum{a_n} \), let \( \displaystyle{L = \lim_{n \to \infty}{\sqrt[n]{a_n}}} \). 
Three cases are possible depending on the value of \(L\). 
When To Use The Root Test
The Root Test is used when you have a function of n that also contains a power with an n. The idea is to remove or change the n in the power. The test itself is fairly straightforward. You just need some practice using it to know under what conditions it is best to use it. One good way to learn when to use this test is by building example pages as described in the Study Techniques section.
Here is a good introduction video.
video by slcmath@pc 

Some Helpful Rules
A powerful rule that will be useful when using the root test is \[ \lim_{n \to \infty}{\ln(f(n))} = \ln\left( \lim_{n \to \infty}{f(n)} \right) \] This is true because the natural log function is continuous. Here are a few additional rules that may be useful.
\(\displaystyle{ \lim_{n \to \infty}{ \sqrt[n]{p}} = 1 }\) 
\(p\) is a positive constant 
\(\displaystyle{ \lim_{n \to \infty}{\sqrt[n]{n^p}} = 1 }\) 
\(p\) is a positive constant 
\(\displaystyle{ \lim_{n \to \infty}{\sqrt[n]{n}} = 1 }\) 
special case of previous line with \(p=1\) 
\(\displaystyle{ \lim_{n \to \infty}{\sqrt[n]{\ln~n}} = 1 }\) 

\(\displaystyle{ \lim_{n \to \infty}{\sqrt[n]{n!}} = \infty }\) 
We prove the first four rules above as practice problems on the L'Hôpital's Rule page. The last one can be proven using Stirlings Formula.
Another helpful formula that sometimes comes up when working these problems is \(\displaystyle{ \lim_{n \to \infty}{ (1+1/n)^n} = e }\). We prove a similar limit on the L'Hopitals Rule page.
Root Test Proof
Here are some videos showing the proof of the Root Test. You do not need to watch these in order to use and understand the Root Test. They are included here for those of you who are interested.
video by slcmath@pc 

video by slcmath@pc 

video by slcmath@pc 

Practice
Unless otherwise instructed, determine whether these series converge or diverge. Use the Root Test, if possible.
Basic 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{4^n} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{4^n} } }\) converges or diverges using the Root Test.
Solution
video by The Organic Chemistry Tutor 

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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{3+5n}{2+3n} \right]^n } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{3+5n}{2+3n} \right]^n } }\) converges or diverges using the Root Test.
Solution
video by The Organic Chemistry Tutor 

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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[\frac{n^2+1}{2n^2+1}\right]^n } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[\frac{n^2+1}{2n^2+1}\right]^n } }\) converges or diverges using the root test.
Final Answer 

The series converges by the root test.
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[\frac{n^2+1}{2n^2+1}\right]^n } }\) converges or diverges using the root test.
Solution
video by Krista King Math 

Final Answer
The series converges by the root test.
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\(\displaystyle{ \sum_{k=1}^{\infty}{ \left[ \frac{k+1}{2k} \right]^k } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{k=1}^{\infty}{ \left[ \frac{k+1}{2k} \right]^k } }\) converges or diverges using the Root Test.
Solution
video by Prof. M.'s Calculus II 

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\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^4}{3^n} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^4}{3^n} } }\) converges or diverges using the Root Test.
Solution
video by slcmath@pc 

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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n+1}{2n1} \right]^n } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n+1}{2n1} \right]^n } }\) converges or diverges using the Root Test.
Solution
video by Trefor Bazett 

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\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(2n+1)^n}{n^{2n}} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(2n+1)^n}{n^{2n}} } }\) converges or diverges using the Root Test.
Solution
video by blackpenredpen 

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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ 1 + \cos(2/n) \right]^n } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ 1 + \cos(2/n) \right]^n } }\) converges or diverges using the Root Test.
Solution
video by slcmath@pc 

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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[3^n e^{n}\right] } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[3^n e^{n}\right] } }\) converges or diverges using the root test.
Final Answer 

The series diverges by the root test.
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[3^n e^{n}\right] } }\) converges or diverges using the root test.
Solution
video by PatrickJMT 

Final Answer
The series diverges by the root test.
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{1}{n^2} + \frac{1}{n} \right]^n } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{1}{n^2} + \frac{1}{n} \right]^n } }\) converges or diverges using the Root Test.
Solution
video by The Organic Chemistry Tutor 

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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{4n}{2n1} \right]^{2n} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{4n}{2n1} \right]^{2n} } }\) converges or diverges using the Root Test.
Solution
video by MIP4U 

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\(\displaystyle{ \sum_{k=1}^{\infty}{ \left[\frac{3^k}{(k+1)^k}\right] } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{k=1}^{\infty}{ \left[\frac{3^k}{(k+1)^k}\right] } }\) converges or diverges using the root test.
Final Answer 

The series converges by the root test.
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{k=1}^{\infty}{ \left[\frac{3^k}{(k+1)^k}\right] } }\) converges or diverges using the root test.
Solution
video by Krista King Math 

Final Answer
The series converges by the root test.
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \sqrt[n]{2} 1 \right]^n } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \sqrt[n]{2} 1 \right]^n } }\) converges or diverges using the Root Test.
Solution
video by blackpenredpen 

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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[\frac{4n^2+1}{5n^28}\right]^n } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[\frac{4n^2+1}{5n^28}\right]^n } }\) converges or diverges using the root test.
Final Answer 

The series converges by the root test.
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[\frac{4n^2+1}{5n^28}\right]^n } }\) converges or diverges using the root test.
Solution
video by MIP4U 

Final Answer
The series converges by the root test.
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^n}{2^{1+4n}} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^n}{2^{1+4n}} } }\) converges or diverges using the Root Test.
Solution
video by The Organic Chemistry Tutor 

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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{e^{3n}}{n^n} \right] } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{e^{3n}}{n^n} \right] } }\) converges or diverges using the root test.
Final Answer 

The series converges by the root test.
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{e^{3n}}{n^n} \right] } }\) converges or diverges using the root test.
Solution
video by MIP4U 

Final Answer
The series converges by the root test.
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{8^{n^2}}{n^n} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{8^{n^2}}{n^n} } }\) converges or diverges using the Root Test.
Solution
video by slcmath@pc 

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\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(1)^n 3^{n+2}}{(n1)^n} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(1)^n 3^{n+2}}{(n1)^n} } }\) converges or diverges using the Root Test.
Solution
video by The Organic Chemistry Tutor 

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Intermediate 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n}{2^n} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n}{2^n} } }\) converges or diverges using the Root Test.
Solution
video by The Organic Chemistry Tutor 

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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n^n}{3^{1+3n}} \right] } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n^n}{3^{1+3n}} \right] } }\) converges or diverges using the root test.
Final Answer 

The series diverges by the root test.
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n^n}{3^{1+3n}} \right] } }\) converges or diverges using the root test.
Solution
video by Krista King Math 

Final Answer
The series diverges by the root test.
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n}{n+1} \right]^{n^2} } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n}{n+1} \right]^{n^2} } }\) converges or diverges using the root test.
Final Answer 

The series converges by the root test.
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n}{n+1} \right]^{n^2} } }\) converges or diverges using the root test.
Solution
video by Krista King Math 

Final Answer
The series converges by the root test.
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ 1  \frac{1}{n} \right]^{n^2} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ 1  \frac{1}{n} \right]^{n^2} } }\) converges or diverges using the Root Test.
Solution
video by blackpenredpen 

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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ 1 + \frac{1}{n} \right]^{n^2} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ 1 + \frac{1}{n} \right]^{n^2} } }\) converges or diverges using the Root Test.
Solution
video by blackpenredpen 

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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{(1)^n(n^n)}{3^{n^3+1}} \right] } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{(1)^n(n^n)}{3^{n^3+1}} \right] } }\) converges or diverges using the root test.
Final Answer 

The series converges by the root test.
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{(1)^n(n^n)}{3^{n^3+1}} \right] } }\) converges or diverges using the root test.
Solution
video by PatrickJMT 

Final Answer
The series converges by the root test.
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{\ln n}{2n} \right]^n } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{\ln n}{2n} \right]^n } }\) converges or diverges using the Root Test.
Solution
video by MIP4U 

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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{(1)^n}{[\tan^{1}(n)]^n} \right] } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{(1)^n}{[\tan^{1}(n)]^n} \right] } }\) converges or diverges using the root test.
Final Answer 

The series converges by the root test.
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{(1)^n}{[\tan^{1}(n)]^n} \right] } }\) converges or diverges using the root test.
Solution
video by PatrickJMT 

Final Answer
The series converges by the root test.
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n^{n1}}{2^{3+n}} \right] } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n^{n1}}{2^{3+n}} \right] } }\) converges or diverges using the root test.
Final Answer 

The series diverges by the Root Test.
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n^{n1}}{2^{3+n}} \right] } }\) converges or diverges using the root test.
Solution
video by MIP4U 

Final Answer
The series diverges by the Root Test.
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