17Calculus - Infinite Series Practice Exam C
17Calculus
This page contains a complete infinite series exam with worked out solutions.
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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]
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Thomas Frank - 10 Study Tips for Earning an A on Your Next Exam [7mins-49secs]
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Recommended Books on Amazon (affiliate links) | ||
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Exam Details | |
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Time | 1 hour |
Questions | 13 |
Total Points | 100 |
Tools | |
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Calculator | not allowed |
Formula Sheet(s) |
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Other Tools | none |
Instructions: |
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- 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.
blackpenredpen - Solving My Calc 2 Exam#3 (Sequence, Infinite Series & Power Series)
video by blackpenredpen |
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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