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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. 
Quick Summary
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 



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 \(\displaystyle{ \int_{k}^{\infty}{f(x)~dx} }\).
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 pseries \( \sum{ 1/n } \).
 Dr Chris Tisdell  Intro to series + the integral test 

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 

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 

Here is a great video giving an intuitive understanding on why this works.
 PatrickJMT  Integral Test for Series: Why It Works 

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 

Instructions   Unless otherwise instructed, determine the convergence or divergence of the following series using the integral test, if possible.
Practice A01 
\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }\) 


The pseries \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }\) converges by the pseries test or the integral test. 
This is a pSeries with \( p=2 > 1\), so the series converges.
We could also have used the Integral Test.
\(\displaystyle{
\begin{array}{rcl}
\int_{1}^{\infty}{\frac{1}{x^2}dx} & = & \lim_{b \to \infty}{\int_{1}^{b}{\frac{1}{x^2}dx}} \\\\
& = & \lim_{b \to \infty}{\int_{1}^{b}{x^{2}dx}} \\\\
& = & \lim_{b \to \infty}{ \left[ x^{1} \right]_{1}^{b}} \\\\
& = & \lim_{b \to \infty}{b^{1} + 1^{1}} \\\\
& = & 0 + 1 = 1
\end{array}
}\)
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.
Practice A01 Final Answer 
The pseries \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }\) converges by the pseries test or the integral test. 
Practice A02 
\(\displaystyle{\sum_{n=2}^{\infty}{\frac{1}{n(\ln n)^2}}}\) 


The series converges by the integral test. 
Although it doesn't affect the final answer, his last step 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.
Practice A02 Final Answer 
The series converges by the integral test. 
Practice A03 
\(\displaystyle{\sum_{n=1}^{\infty}{\frac{n}{(n^2+1)^2}}}\) 


The series converges by the integral test. 
Practice A03 Final Answer 
The series converges by the integral test. 
Practice A04 
\(\displaystyle{\sum_{n=1}^{\infty}{ne^{n^2}}}\) 


The series converges by the integral test. 
Practice A04 Final Answer 
The series converges by the integral test. 
Practice A05 
\(\displaystyle{\sum_{n=1}^{\infty}{\frac{n}{n^2+1}}}\) 


The series diverges by the integral test. 
Practice A05 Final Answer 
The series diverges by the integral test. 
Practice A06 
\(\displaystyle{\sum_{n=1}^{\infty}{\frac{1}{n^2+1}}}\) 


The series converges by the integral test. 
Practice A06 Final Answer 
The series converges by the integral test. 
Practice A07 
\(\displaystyle{\sum_{n=1}^{\infty}{\frac{1}{\sqrt{n}}}}\) 


The series diverges by the integral test. 
Practice A07 Final Answer 
The series diverges by the integral test. 
Practice A08 
\(\displaystyle{\sum_{n=1}^{\infty}{\frac{1}{n}}}\) 


The series diverges by the integral test. 
Practice A08 Final Answer 
The series diverges by the integral test. 
Practice A09 
\(\displaystyle{\sum_{n=1}^{\infty}{\frac{1}{n^{1.1}}}}\) 


The series converges by the integral test. 
Practice A09 Final Answer 
The series converges by the integral test. 
Practice A10 
\(\displaystyle{\sum_{n=1}^{\infty}{\frac{1}{5n2}}}\) 

Practice B01 
\(\displaystyle{\sum_{n=1}^{\infty}{\frac{1}{n^24n+5}}}\) 


The series converges by the integral test. 
Although this problem would be more easily solved using one of the comparison tests, this is a great example using the integral test.
Practice B01 Final Answer 
The series converges by the integral test. 
Practice B02 
\(\displaystyle{\sum_{n=1}^{\infty}{\frac{1}{3^n}}}\) 


The series converges by the integral test. 
Practice B02 Final Answer 
The series converges by the integral test. 
Practice B03 
\(\displaystyle{\sum_{n=2}^{\infty}{\frac{1}{n[(\ln n)^2+4]}}}\) 


The series converges by the integral test. 
Practice B03 Final Answer 
The series converges by the integral test. 
Practice B04 
\(\displaystyle{\sum_{n=1}^{\infty}{\left[\frac{4}{n\sqrt[3]{n}}+\frac{5}{n}\right]}}\) 


The series diverges by the integral test. 
Practice B04 Final Answer 
The series diverges by the integral test. 
Practice B05 
\(\displaystyle{\sum_{n=1}^{\infty}{\frac{\ln(n)}{n^2}}}\) 


The series converges by the integral test. 
Practice B05 Final Answer 
The series converges by the integral test. 