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 limits infinite limits l'hôpital's rule infinite series basics

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

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17calculus > infinite series > root test

 Root Test When To Use The Root Test Some Helpful Rules Practice

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, it can be used to determine convergence or divergence much easier than using other techniques in these very specific cases.

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.

A powerful rule that will be useful when using the root test is $$\displaystyle{ \lim_{n \to \infty}{\ln(f(n))} = \ln\left( \lim_{n \to \infty}{f(n)} \right) }$$
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.

### Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

GOT IT. THANKS!

Instructions - - Unless otherwise instructed, determine whether these series converge or diverge using the root test.

Basic Problems

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

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[\frac{n^2+1}{2n^2+1}\right]^n } }$$

The series converges by the root test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[\frac{n^2+1}{2n^2+1}\right]^n } }$$

Solution

### 264 solution video

video by PatrickJMT

The series converges by the root test.

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

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[3^n e^{-n}\right] } }$$

The series diverges by the root test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[3^n e^{-n}\right] } }$$

Solution

### 267 solution video

video by PatrickJMT

The series diverges by the root test.

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

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{k=1}^{\infty}{ \left[\frac{3^k}{(k+1)^k}\right] } }$$

The series converges by the root test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{k=1}^{\infty}{ \left[\frac{3^k}{(k+1)^k}\right] } }$$

Solution

### 270 solution video

video by Krista King Math

The series converges by the root test.

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

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[\frac{4n^2+1}{5n^2-8}\right]^n } }$$

The series converges by the root test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[\frac{4n^2+1}{5n^2-8}\right]^n } }$$

Solution

### 271 solution video

video by MIP4U

The series converges by the root test.

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

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{e^{3n}}{n^n} \right] } }$$

The series converges by the root test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{e^{3n}}{n^n} \right] } }$$

Solution

### 273 solution video

video by MIP4U

The series converges by the root test.

Intermediate Problems

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

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n^n}{3^{1+3n}} \right] } }$$

The series diverges by the root test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n^n}{3^{1+3n}} \right] } }$$

Solution

### 265 solution video

video by PatrickJMT

The series diverges by the root test.

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

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n}{n+1} \right]^{n^2} } }$$

The series converges by the root test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n}{n+1} \right]^{n^2} } }$$

Solution

### 266 solution video

video by PatrickJMT

The series converges by the root test.

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

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{(-1)^n(n^n)}{3^{n^3+1}} \right] } }$$

The series converges by the root test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{(-1)^n(n^n)}{3^{n^3+1}} \right] } }$$

Solution

### 268 solution video

video by PatrickJMT

The series converges by the root test.

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

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{(-1)^n}{[\tan^{-1}(n)]^n} \right] } }$$

The series converges by the root test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{(-1)^n}{[\tan^{-1}(n)]^n} \right] } }$$

Solution

### 269 solution video

video by PatrickJMT

The series converges by the root test.

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

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n^{n-1}}{2^{3+n}} \right] } }$$

The series diverges by the root test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{n^{n-1}}{2^{3+n}} \right] } }$$

Solution

video by MIP4U