\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus - Infinite Series Practice Exam C

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

This page contains a complete infinite series exam with worked out solutions.

Practice Exam Tips

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.

Thomas Frank - 5 Rules (and One Secret Weapon) for Acing Multiple Choice Tests [9mins-42secs]

video by Thomas Frank

Thomas Frank - 10 Study Tips for Earning an A on Your Next Exam [7mins-49secs]

video by Thomas Frank

Exam Details

Time

1 hour

Questions

13

Total Points

100

Tools

Calculator

not allowed

Formula Sheet(s)

table of series tests

Other Tools

none

Instructions:

- Show all your work.
- For each problem, correct answers are worth 10%. The remaining points are earned by showing calculations and giving reasoning that justify your conclusions.
- Correct notation counts (i.e. points will be taken off for incorrect notation).
- Give exact, simplified answers.

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.

blackpenredpen - Solving My Calc 2 Exam#3 (Sequence, Infinite Series & Power Series)

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

blackpenredpen - 4099 video 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

blackpenredpen - 4100 video 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

blackpenredpen - 4101 video 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

blackpenredpen - 4102 video solution

video by blackpenredpen

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Find the radius of convergence for the power series \(\displaystyle{ \sum_{n=1}^{\infty}{ [2^n(x-3)^n] } }\)

Problem Statement

Find the radius of convergence for the power series \(\displaystyle{ \sum_{n=1}^{\infty}{ [2^n(x-3)^n] } }\)

Solution

blackpenredpen - 4103 video solution

video by blackpenredpen

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Consider the recursive sequence where \( a_1 = 5 \), \( a_n = 8 - a_{n-1} \) 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 \( a_1 = 5 \), \( a_n = 8 - a_{n-1} \) for \( n \geq 2 \). Is the sequence increasing or decreasing? Does the sequence converge or diverge? If it converges, find the convergence value.

Solution

blackpenredpen - 4104 video solution

video by blackpenredpen

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Determine the first four non-zero terms of the power series for \( \ln x \) at \( x = 2 \).

Problem Statement

Determine the first four non-zero terms of the power series for \( \ln x \) at \( x = 2 \).

Solution

blackpenredpen - 4105 video solution

video by blackpenredpen

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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

blackpenredpen - 4106 video 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

blackpenredpen - 4107 video 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

blackpenredpen - 4108 video 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

blackpenredpen - 4109 video 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

blackpenredpen - 4110 video solution

video by blackpenredpen

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Determine whether \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2+2n+4}{\sqrt{n^5+8n^2-2}} } }\) converges or diverges. Justify your answer.

Problem Statement

Determine whether \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2+2n+4}{\sqrt{n^5+8n^2-2}} } }\) converges or diverges. Justify your answer.

Solution

blackpenredpen - 4111 video 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

blackpenredpen - 4112 video 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

blackpenredpen - 4113 video 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

blackpenredpen - 4114 video 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

blackpenredpen - 4115 video solution

video by blackpenredpen

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