This page contains a complete infinite series exam with worked out solutions.
Each exam page contains a full exam with detailed solutions. Most of these are actual exams from previous semesters used in college courses. You may use these as practice problems or as practice exams. Here are some suggestions on how to use these to help you prepare for your exams.
 Set aside a chunk of full, uninterrupted time, usually an hour to two, to work each exam.
 Go to a quiet place where you will not be interrupted that duplicates your exam situation as closely as possible.
 Use the same materials that you are allowed in your exam (unless the instructions with these exams are more strict).
 Use your calculator as little as possible except for graphing and checking your calculations.
 Work the entire exam before checking any solutions.
 After checking your work, rework any problems you missed and go to the 17calculus page discussing the material to perfect your skills.
 Work as many practice exams as you have time for. This will give you practice in important techniques, experience in different types of exam problems that you may see on your own exam and help you understand the material better by showing you what you need to study.
IMPORTANT 
Exams can cover only so much material. Instructors will sometimes change exams from one semester to the next to adapt an exam to each class depending on how the class performs during the semester while they are learning the material. So just because you do well (or not) on these practice exams, does not necessarily mean you will do the same on your exam. Your instructor may ask completely different questions from these. That is why working lots of practice problems will prepare you better than working just one or two practice exams.
Calculus is not something that can be learned by reading. You have to work problems on your own and struggle through the material to really know calculus, do well on your exam and be able to use it in the future.
Okay, so here are a few videos we recommend that expand on the some of the above tips and also provide some insight on taking exams. This guy has lots of other videos about how to succeed in college, so we recommend his YouTube channel, Thomas Frank.
video by Thomas Frank 

video by Thomas Frank 

Recommended Books on Amazon (affiliate links)  

Exam Details  

Time  1 hour 
Questions  13 
Total Points  100 
Tools  

Calculator  not allowed 
Formula Sheet(s) 

Other Tools  none 
Instructions: 

 Show all your work. 
This exam is from the youtube channel, black pen  red pen. Below is the entire video and here is the link to the list of problems. His exam is multiple choice. We recommend that you do not look at the choices when working these problems. We show the problems below without the choices to help you better prepare for your exam.
video by blackpenredpen 

Solve these problems making sure to show your work. As part of your final answer, make sure you specify which test(s) definitively proves your conclusion.
Does \(\displaystyle{ \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10} + . . . }\) converge or diverge? If it converges, what is the convergence value?
Problem Statement
Does \(\displaystyle{ \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10} + . . . }\) converge or diverge? If it converges, what is the convergence value?
Solution
video by blackpenredpen 

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Which of the following series converges absolutely? Show your work.
A. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(1)^n}{\sqrt[3]{n}} } }\)  B. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(1)^n}{2^n} } }\)  C. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(1)^n n^2}{n^3+1} } }\)  
D. \(\displaystyle{ \sum_{n=1}^{\infty}{ (1)^n } }\)  E. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(1)^n 3^n}{2^n} } }\) 
Problem Statement
Which of the following series converges absolutely? Show your work.
A. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(1)^n}{\sqrt[3]{n}} } }\)  B. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(1)^n}{2^n} } }\)  C. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(1)^n n^2}{n^3+1} } }\)  
D. \(\displaystyle{ \sum_{n=1}^{\infty}{ (1)^n } }\)  E. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(1)^n 3^n}{2^n} } }\) 
Solution
video by blackpenredpen 

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If possible, evaluate \(\displaystyle{ \sum_{n=1}^{\infty}{ \left( e^{1/\sqrt{n}}  e^{1/\sqrt{n+1}} \right) } }\) or show that the series diverges.
Problem Statement
If possible, evaluate \(\displaystyle{ \sum_{n=1}^{\infty}{ \left( e^{1/\sqrt{n}}  e^{1/\sqrt{n+1}} \right) } }\) or show that the series diverges.
Solution
video by blackpenredpen 

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Which of these series diverges by the Divergence Test? Show your work.
A. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n} \ln n} } }\)  B. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\tan^{1}(n)}{n^2} } }\)  C. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2+2n}{n^3+4n+9} } }\)  
D. \(\displaystyle{ \sum_{n=1}^{\infty}{ \cos(1/n) } }\)  E. \(\displaystyle{ \sum_{n=1}^{\infty}{ \sin(1/n) } }\) 
Problem Statement
Which of these series diverges by the Divergence Test? Show your work.
A. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n} \ln n} } }\)  B. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\tan^{1}(n)}{n^2} } }\)  C. \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2+2n}{n^3+4n+9} } }\)  
D. \(\displaystyle{ \sum_{n=1}^{\infty}{ \cos(1/n) } }\)  E. \(\displaystyle{ \sum_{n=1}^{\infty}{ \sin(1/n) } }\) 
Solution
video by blackpenredpen 

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Find the radius of convergence for the power series \(\displaystyle{ \sum_{n=1}^{\infty}{ [2^n(x3)^n] } }\)
Problem Statement
Find the radius of convergence for the power series \(\displaystyle{ \sum_{n=1}^{\infty}{ [2^n(x3)^n] } }\)
Solution
video by blackpenredpen 

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Consider the recursive sequence where \( a_1 = 5 \), \( a_n = 8  a_{n1} \) for \( n \geq 2 \). Is the sequence increasing or decreasing? Does the sequence converge or diverge? If it converges, find the convergence value.
Problem Statement
Consider the recursive sequence where \( a_1 = 5 \), \( a_n = 8  a_{n1} \) for \( n \geq 2 \). Is the sequence increasing or decreasing? Does the sequence converge or diverge? If it converges, find the convergence value.
Solution
video by blackpenredpen 

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Determine the first four nonzero terms of the power series for \( \ln x \) at \( x = 2 \).
Problem Statement
Determine the first four nonzero terms of the power series for \( \ln x \) at \( x = 2 \).
Solution
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Integrate \( \int{ e^{x^2} ~ dx } \) as a power series and find the radius of convergence.
Problem Statement
Integrate \( \int{ e^{x^2} ~ dx } \) as a power series and find the radius of convergence.
Solution
video by blackpenredpen 

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Integrate \(\displaystyle{ \int{ \frac{1}{1+8x^3} ~ dx } }\) as a power series and find the radius of convergence.
Problem Statement
Integrate \(\displaystyle{ \int{ \frac{1}{1+8x^3} ~ dx } }\) as a power series and find the radius of convergence.
Solution
video by blackpenredpen 

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Determine whether \(\displaystyle{ \sum_{n=1}^{\infty}{ \sin^3(1/n) } }\) converges or diverges. Justify your answer.
Problem Statement
Determine whether \(\displaystyle{ \sum_{n=1}^{\infty}{ \sin^3(1/n) } }\) converges or diverges. Justify your answer.
Solution
video by blackpenredpen 

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There is no question 10.
Determine whether \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n!}} } }\) converges or diverges. Justify your answer.
Problem Statement
Determine whether \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n!}} } }\) converges or diverges. Justify your answer.
Solution
video by blackpenredpen 

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Let \(\displaystyle{ a_n = 2 \left( \frac{3}{4} \right)^n }\)
(a) Does \( \{ a_n \} \) converge? If so, to what value?
(b) Does \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }\) converge? If so, to what value?
Problem Statement
Let \(\displaystyle{ a_n = 2 \left( \frac{3}{4} \right)^n }\)
(a) Does \( \{ a_n \} \) converge? If so, to what value?
(b) Does \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }\) converge? If so, to what value?
Solution
video by blackpenredpen 

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Determine whether \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2+2n+4}{\sqrt{n^5+8n^22}} } }\) converges or diverges. Justify your answer.
Problem Statement
Determine whether \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2+2n+4}{\sqrt{n^5+8n^22}} } }\) converges or diverges. Justify your answer.
Solution
video by blackpenredpen 

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Give an example of \(a_n\) so that \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } = 2 }\)
Problem Statement
Give an example of \(a_n\) so that \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } = 2 }\)
Solution
video by blackpenredpen 

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Give an example of \(a_n\) and \(b_n\) so that both \(\displaystyle{ \sum_{n=1}^{\infty}{a_n} }\) and \(\displaystyle{ \sum_{n=1}^{\infty}{b_n} }\) diverge but \(\displaystyle{ \sum_{n=1}^{\infty}{ (a_n b_n) } }\) converges.
Problem Statement
Give an example of \(a_n\) and \(b_n\) so that both \(\displaystyle{ \sum_{n=1}^{\infty}{a_n} }\) and \(\displaystyle{ \sum_{n=1}^{\infty}{b_n} }\) diverge but \(\displaystyle{ \sum_{n=1}^{\infty}{ (a_n b_n) } }\) converges.
Solution
video by blackpenredpen 

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Give an example of \(a_n\) so that \(a_n \neq 0 \) for all \(n\) but \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } = 0 }\)
Problem Statement
Give an example of \(a_n\) so that \(a_n \neq 0 \) for all \(n\) but \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } = 0 }\)
Solution
video by blackpenredpen 

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Give an example of \(a_n\) and \(b_n\) so that \(a_n \neq b_n \) for all \(n\) but \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } = \sum_{n=1}^{\infty}{ b_n } }\)
Problem Statement
Give an example of \(a_n\) and \(b_n\) so that \(a_n \neq b_n \) for all \(n\) but \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } = \sum_{n=1}^{\infty}{ b_n } }\)
Solution
video by blackpenredpen 

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