17Calculus Infinite Series - Root Test

17Calculus

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.

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 $$L$$.
$$L < 1$$: The series converges absolutely.
$$L = 1$$: The Root Test is inconclusive.
$$L> 1$$: The series diverges.

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 straight-forward. 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.

slcmath@pc - The Root Test - Introduction

video by slcmath@pc

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.

slcmath@pc - The Root Test - Proof of Part (a)

video by slcmath@pc

slcmath@pc - The Root Test - Proof of Part (b)

video by slcmath@pc

slcmath@pc - The Root Test - Proof of Part (c)

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

The Organic Chemistry Tutor - 3240 video solution

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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

The Organic Chemistry Tutor - 3241 video solution

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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.

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

Krista King Math - 264 video solution

video by Krista King Math

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

Prof. M.'s Calculus II - 3252 video solution

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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

slcmath@pc - 3254 video solution

video by slcmath@pc

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n+1}{2n-1} \right]^n } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n+1}{2n-1} \right]^n } }$$ converges or diverges using the Root Test.

Solution

Trefor Bazett - 3246 video 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

blackpenredpen - 3248 video 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

slcmath@pc - 3256 video 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.

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

PatrickJMT - 267 video solution

video by PatrickJMT

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

The Organic Chemistry Tutor - 3242 video solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{4n}{2n-1} \right]^{2n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{4n}{2n-1} \right]^{2n} } }$$ converges or diverges using the Root Test.

Solution

MIP4U - 3253 video 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.

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

Krista King Math - 270 video solution

video by Krista King Math

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

blackpenredpen - 3249 video solution

video by blackpenredpen

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \left[\frac{4n^2+1}{5n^2-8}\right]^n } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[\frac{4n^2+1}{5n^2-8}\right]^n } }$$ converges or diverges using the root test.

The series converges by the root test.

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[\frac{4n^2+1}{5n^2-8}\right]^n } }$$ converges or diverges using the root test.

Solution

MIP4U - 271 video solution

video by MIP4U

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

The Organic Chemistry Tutor - 3243 video solution

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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.

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

MIP4U - 273 video solution

video by MIP4U

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

slcmath@pc - 3255 video solution

video by slcmath@pc

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n 3^{n+2}}{(n-1)^n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n 3^{n+2}}{(n-1)^n} } }$$ converges or diverges using the Root Test.

Solution

The Organic Chemistry Tutor - 3245 video solution

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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

The Organic Chemistry Tutor - 3244 video solution

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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.

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

Krista King Math - 265 video solution

video by Krista King Math

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.

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

Krista King Math - 266 video solution

video by Krista King Math

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

blackpenredpen - 3247 video 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

blackpenredpen - 3250 video 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.

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

PatrickJMT - 268 video solution

video by PatrickJMT

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

MIP4U - 3251 video 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.

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

PatrickJMT - 269 video solution

video by PatrickJMT

The series converges by the root test.

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n^{n-1}}{2^{3+n}} \right] } }$$

Problem Statement

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

The series diverges by the Root Test.

Problem Statement

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

Solution

MIP4U - 272 video solution

video by MIP4U

The series diverges by the Root Test.

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