You CAN Ace Calculus

### Topics You Need To Understand For This Page

 limits at infinity integration improper integrals infinite series basics

### Related Topics and Links

integral test youtube playlist

WikiBooks - Integral Test

### Integral Test Quick Notes

 used to prove convergence yes used to prove divergence yes can be inconclusive yes $$a_n$$ must be positive $$a_n$$ must be decreasing requires that the integrand must be integrable (not always possible) requires the evaluation of infinite limits (after integration) if the result of the limit (after integration) does not exist (different than diverges), this test is inconclusive

### 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

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

The Integral Test is easy to use and is good to use when the ratio test and the comparison tests won't work and you are pretty sure that you can evaluate the integral. The idea of this test is to evaluate the improper integral $$\displaystyle{ \int_{k}^{\infty}{f(x)~dx} }$$.

Integral Test

For a series $$\displaystyle{\sum_{n=1}^{\infty}{a_n}}$$ where we can find a positive, continuous and decreasing function $$f$$ for $$n > k$$ and $$a_n = f(n)$$, then we know that if $\int_{k}^{\infty}{f(x) ~ dx}$ converges, the series also converges. Similarly when the integral diverges, the series also diverges.

 Two Things To Watch For

1. The value of k
First, you need to find a constant k such that the function satisfies all of these conditions for all $$n > k$$:

 continuous positive decreasing

One of the favorite tricks that teachers like to put on exams (which I fell for when I first took the class) is to tell you to use the Integral Test but then not give you k. Many books just show this integral with $$k=1$$, which is not always valid. So be careful.
How To Find k:
The best way is to calculate the critical values of the function and then check that the derivative is negative to the right of the largest critical value. Then, if you have access to a graphing calculator, do a quick plot to check your answer. If everything looks good, choose k to be greater than the largest critical value. Any value will do, so choose one that will be easy to use in the integration.
There is no one value that will always work. It depends on the function.

2. The final value of the integration
Secondly, if you get a finite value for the integral and determine that the series converges, the finite value you got from the integral is NOT what the series converges to. The number itself has no meaning in this context (ie. we don't use the value of the number to tell us anything about the series). The significance of it lies in whether it is finite or not. That's it. That's all the information you can get from that number. So do NOT assume that the series converges to that number.

Okay, let's watch some videos to see how this test works.

In this first video clip, he does a great job explaining the integral test. He uses the integral test to show the divergence of the p-series $$\sum{ 1/n }$$.

### Dr Chris Tisdell - Intro to series + the integral test [11min-23secs]

video by Dr Chris Tisdell

In this next video, the instructor explains the integral test in more detail by using it on the two series $$\sum{ 1/n }$$ and $$\sum{ 1/n^2 }$$ to show that one diverges and the other converges.

### Dr Chris Tisdell - Integral test for Series [13min-43secs]

video by Dr Chris Tisdell

Here is another good explanation of the integral test. He looks at the sum $$\displaystyle{ \sum_{n=1}^{\infty}{\frac{1}{n^p}} }$$.

### PatrickJMT - Integral Test - Basic Idea [3min-26secs]

video by PatrickJMT

Here is a great video giving an intuitive understanding on why this works.

### PatrickJMT - Integral Test for Series: Why It Works [14min-46secs]

video by PatrickJMT

This last video discusses the remainder estimate for the integral test. Although not required to understand how to use the integral test, this video will help you understand more intuitively what is going on.

### PatrickJMT - Remainder Estimate for the Integral Test [7min-45secs]

video by PatrickJMT

### 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 the convergence or divergence of the following series using the integral test, if possible.

Basic Problems

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }$$

The series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }$$ converges by the p-series test or the integral test.

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }$$

Solution

This is a p-series with $$p=2 > 1$$, so the series converges by the p-series test.
We could also have used the Integral Test, as follows.

 $$\displaystyle{ \int_{1}^{\infty}{\frac{1}{x^2}dx} }$$ $$\displaystyle{ \lim_{b \to \infty}{\int_{1}^{b}{\frac{1}{x^2}dx}} }$$ $$\displaystyle{ \lim_{b \to \infty}{\int_{1}^{b}{x^{-2}dx}} }$$ $$\displaystyle{ \lim_{b \to \infty}{ \left[ -x^{-1} \right]_{1}^{b}} }$$ $$\displaystyle{ \lim_{b \to \infty}{-b^{-1} + 1^{-1}} }$$ $$0 + 1 = 1$$

Since the improper integral is finite, the series converges by the Integral Test.
Note: The value $$1$$, from the integral is NOT necessarily what the series converges to. The significance of this number is only that it is finite.

The series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }$$ converges by the p-series test or the integral test.

$$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{n(\ln n)^2} } }$$

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{n(\ln n)^2} } }$$

The series converges by the integral test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{n(\ln n)^2} } }$$

Solution

Although it doesn't affect the final answer, his last step in the video should be $$\displaystyle{ 0 - \left( \frac{-1}{\ln 2} \right) = \frac{1}{\ln 2}}$$. He writes $$\ln 2$$ instead of $$1/(\ln 2)$$. Since $$1/(\ln 2)$$ is finite, the series converges by the integral test.

### 240 solution video

video by MIT OCW

The series converges by the integral test.

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

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n}{(n^2+1)^2} } }$$

The series converges by the integral test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n}{(n^2+1)^2} } }$$

Solution

### 241 solution video

video by Dr Chris Tisdell

The series converges by the integral test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ ne^{-n^2} } }$$

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ ne^{-n^2} } }$$

The series converges by the integral test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ ne^{-n^2} } }$$

Solution

### 242 solution video

video by PatrickJMT

The series converges by the integral test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n}{n^2+1} } }$$

Problem Statement

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

The series diverges by the integral test.

Problem Statement

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

Solution

### 243 solution video

video by PatrickJMT

The series diverges by the integral test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2+1} } }$$

Problem Statement

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

The series converges by the integral test.

Problem Statement

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

Solution

### 248 solution video

video by Krista King Math

The series converges by the integral test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n}} } }$$

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n}} } }$$

The series diverges by the integral test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n}} } }$$

Solution

### 249 solution video

video by Educator.com

The series diverges by the integral test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n} } }$$

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n} } }$$

The series diverges by the integral test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n} } }$$

Solution

### 250 solution video

video by MIP4U

The series diverges by the integral test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{1.1}} } }$$

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{1.1}} } }$$

The series converges by the integral test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{1.1}} } }$$

Solution

### 251 solution video

video by MIP4U

The series converges by the integral test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{5n-2} } }$$

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{5n-2} } }$$

Solution

### 1213 solution video

video by MIP4U

Intermediate Problems

$$\displaystyle{\sum_{n=1}^{\infty}{\frac{1}{n^2-4n+5}}}$$

Problem Statement

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

The series converges by the integral test.

Problem Statement

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

Solution

Although this problem would be more easily solved using one of the comparison tests, this is a great example using the integral test.

### 244 solution video

video by PatrickJMT

The series converges by the integral test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{3^n} } }$$

Problem Statement

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

The series converges by the integral test.

Problem Statement

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

Solution

### 245 solution video

video by PatrickJMT

The series converges by the integral test.

$$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{n[(\ln n)^2+4]} } }$$

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{n[(\ln n)^2+4]} } }$$

The series converges by the integral test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{n[(\ln n)^2+4]} } }$$

Solution

### 246 solution video

video by PatrickJMT

The series converges by the integral test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \left[\frac{4}{n\sqrt[3]{n}}+\frac{5}{n}\right] } }$$

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[\frac{4}{n\sqrt[3]{n}}+\frac{5}{n}\right] } }$$

The series diverges by the integral test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[\frac{4}{n\sqrt[3]{n}}+\frac{5}{n}\right] } }$$

Solution

### 247 solution video

video by PatrickJMT

The series diverges by the integral test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\ln(n)}{n^2} } }$$

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\ln(n)}{n^2} } }$$

The series converges by the integral test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\ln(n)}{n^2} } }$$

Solution

### 252 solution video

video by MIP4U

The series converges by the integral test.

$$\displaystyle{ \sum_{k=2}^{\infty}{ \frac{2}{k\ln k} } }$$

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{k=2}^{\infty}{ \frac{2}{k\ln k} } }$$

The series diverges by the integral test.

Problem Statement

Determine convergence or divergence of the series $$\displaystyle{ \sum_{k=2}^{\infty}{ \frac{2}{k\ln k} } }$$

Solution

### 2443 solution video

video by Math Cabin