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

17Calculus Infinite Series - Practice Problems

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This page contains a list of infinite series practice problems. These problems are not in any special order, similar to how you might find them on an exam. We work these problems multiple times with different tests to compare which might be best and explain why.

There are hundreds more practice problems on pages dedicated to specific infinite series techniques. Select a topic from the menu to go to a specific topic page.

Determining convergence or divergence of a series can often be done using several different tests. There is usually not just one way to work an infinite series problem. However, there is usually a best way. The practice problems listed here have extremely detailed solutions showing many possible ways to determine convergence or divergence. This allows you to compare the methods and begin to see patterns. The best way to learn to solve these problems efficiently is to start with the guidelines below and then tweak them as you get experience. Don't be afraid to make mistakes or go down rabbit trails to dead-ends. These are valuable learning experiences that will help you prepare for exams.

Infinite Series Study Techniques

Learning Infinite Series tests and how to apply them can be the most difficult topic you will come across in your first year of calculus. However, there are some study techniques that will help you. Here are some suggestions.

1. Print out the tests table. Write in comments of your own that will help you remember them. Use this table while doing your homework. Looking at the table enough will commit it to memory without having to memorize it.

2. Set aside several sheets of paper with the name of a different series test (and any other information you think you might need) at the top of each sheet. Whenever you are given a series to test for convergence or divergence, write that series on the sheet with the test you used and whether it converged or diverged. If you used one of the comparison tests, write the series you used for comparison next to the series you analyzed. As you build these sheets, you will start to see patterns. Once you see the patterns, you will know by looking at a series, what test might work. This is an extremely powerful technique.

3. In your textbook, go through the exercises at the end of each section and look for patterns. For example, if the section is on the Limit Comparison Test, go through the exercises that tell you to use the Limit Comparison Test and see if you can pick up any patterns in the problems. For example, are there a lot of factorials or do most of the problems contain polynomials, logarithms or exponentials.

4. Work as many practice exercises as you can. I know you are busy and it's difficult sometimes to work exercises that you don't get credit for. However, if you work extra exercises and build the pages as I recommended in #2 above, you could very possibly ace the exam. Is that enough incentive?

5. As you are working practice exercises, work the same problem multiple times trying different tests. Convergence or divergence of many, if not most, of the series can be proven in multiple ways using different tests. This technique will help you understand which technique is best in terms of speed and ease. You will also learn which tests not to use on certain types of problems. This is a critical technique. If you do not do this, you will probably run across a problem that seemed to work before but, because of some little twist in the series, won't work. And if that happens, you will be stuck and not have a backup plan. A third benefit is that, if you get different answers from using different techniques, you will know you are doing something wrong BEFORE the exam and be able to correct it.

6. We have never found one textbook that will explain everything you need to know to understand infinite series and ace your exams. So we recommend these inexpensive books.
Start with a good textbook. We recommend Larson Calculus. An older edition will do. We have used editions ETF 3, 4 and 5 and find them all very good. So choose the cheapest, you may be able to find a used one for around 10USD.

Larson Calculus ETF 3rd Edition
Larson Calculus ETF 4th Edition

Next choose a supplemental book dedicated to only infinite series. We recommend one of these. Read about them carefully since some of them are very theory intensive and others are application intensive. Which one you choose will depend how deeply your class requires your knowledge. Math majors should choose the one that has a lot of theory. Engineers should focus on the applications.

Infinite Series (Dover Books on Mathematics) by James M Hyslop
Infinite Series (Dover Books on Mathematics) by Isidore Isaac Hirschman
Theory and Application of Infinite Series (Dover Books on Mathematics) by Konrad Knopp

If you use these study techniques and tools, you may find that working infinite series problems is actually pretty easy. I know, at this time you don't think so but as you master the concepts, the simplicity of these problems will come through. Right now, they seem complicated because all the different techniques are jumbled around in your head. But that doesn't mean you can't understand them with the right tools and a little bit of work. You now have the tools. Are you willing to put in the work?

Where To Start - Choosing A Test To Determine Convergence or Divergence

As you are first learning these tests, here is a hint on how to determine convergence or divergence of a infinite series. Use the tests in this order until one of them tells you that the series converges or diverges:

Group 1. Divergence (nth-Term) Test
Remember that the divergence test can test only for divergence. If the limit goes to zero, you don't know if the series converges or diverges. Said another way, the only two conclusions you can reach from the divergence test are, divergence or inconclusive. If you don't determine divergence, you have to use another test.

Group 2. See if the series matches one of these special forms.
- p-Series
- Geometric Series
- Alternating Series
- Telescoping Series

Group 3. Core Series Tests
- Ratio Test
- Limit Comparison Test or Direct Comparison Test (whichever you find easiest to use)
- Integral Test
- Root Test

The series table is organized in this format.

Once you get some experience, you will find that you can look at a series and determine by what terms are involved, where to start. But this comes only after working many practice problems and deliberate work on your part to discerning patterns. Here are just a few guidelines that may help ( in no special order ). Add more to this list as you learn them.
1. If the series contains a factorial, try the Ratio Test or the Limit Comparison Test.
2. If the series contains a term with n that also has a power of n, try the Root Test.
3. Start with the Alternating Series if the series is alternating.
4. If the series has positive and negative terms, try determining the convergence or divergence of the absolute value of the series and then apply the absolute convergence theorem.
5. If the series looks integrable, try the Integral Test.
6. If you have a polynomial in the denominator that can be factored and no complicated terms in the numerator, try expanding into partial fractions and see if you have a Telescoping Series.

Of course, these guidelines can be used only if you have the option of choosing any test. Make sure you read the problem statement carefully and, if there is any confusion, ask your instructor what they expect.

Here is a video that builds on this discussion. Basically, he has a list of series and he explains his thought process in choosing a test. He doesn't actually work any problems in this video but he does give a great overview on how to look at each series. You will need some actual experience working problems and have some knowledge of all the tests to be able to understand this video.

PatrickJMT - Infinite Series [12min-46secs]

video by PatrickJMT

Infinite Series Exam Preparation

This section gives you techniques and advice on how to prepare for your infinite series exam (or for the infinite series part of your exam). Most of this material can be found spread around on other pages (we have included links to those locations) but this section brings together everything you need to prepare for your exam. Let's get started.

1. Basic Preparation
- Read through all sections on the main infinite series page and use those techniques as you learn.
- Study your textbook and each page on this site dealing with specific techniques as you come across them in your textbook.
- Work assigned practice problems for class and the practice problems on this site at the bottom of the page for each specific technique.

2. Specific Preparation
- Have your table of tests handy. If you find that the infinite series table on this site is difficult for you to use, rewrite it so that you understand it.
- If you built a set of pages with problems that you practiced (as described above in the study techniques section), then have those handy while you are preparing. If you find any other problems that are not listed on these pages, then add them as you study. However, try not use these pages until you are completely stuck and don't have a clue what to do.
- Work the practice problems on this page, as many as you have time for . . . the more, the better. These practice problems are not arranged in any order, just like you might find on your exam. So you don't have any clues on which test to use. If you need a reminder about which test to choose, read the section above.

3. Final Touches
Okay, now you think you are ready for your exam. To make sure, go to the infinite series exams and work through the exams just as you would for your exam. Try to set up the same conditions as you will have in your own exam. This includes what materials are allowed on the exam, how much time you have, how much noise is in the room, etc.
This is an extremely important part of the preparation. It will remove some anxiety and make it easier to remember what you know.

Okay, now you are ready to ace your exam. Go into your exam with confidence, knowing you can do it.

Practice Problems

There are hundreds of practice problems scattered around on the various infinite series pages. Since infinite series problems can usually be solved several ways, the problems below combine all the solutions so that you can compare techniques. To study a specific technique, select a topic from the menu to go to the page discussing that technique.

Instructions:
1. Determine the convergence or divergence of the following series.
2. If possible, determine the value the series converges to.
3. Determine if a convergent series converges absolutely or conditionally.

Caveat - This list of practice problems is in its infancy. After looking at one or two solutions, you can tell that these take significant time to develop and layout. We plan to add more of these in the future but for now, their value is to demonstrate a suggested way to determine which test to use.

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

\(\displaystyle{\sum_{n=0}^{\infty}{\frac{1}{3^n+n}}}\)

\(\displaystyle{\sum_{n=1}^{\infty}{\frac{n}{n^2-\cos^2(n)}}}\)

\(\displaystyle{ \sum_{n=1}^{\infty}{\frac{n^2-1}{n^3+4}}}\)

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{3/2}+1} } }\)

\(\displaystyle{\sum_{n=1}^{\infty}{\frac{\sin(n)}{n^3+n+1}}}\)

\(\displaystyle{\sum_{n=1}^{\infty}{\frac{n~\cos(n\pi)}{2n-1}}}\)

\(\displaystyle{ \sum_{k=1}^{\infty}{ \left[ \frac{1}{k+1} - \frac{1}{k+2} \right] } }\)

You CAN Ace Calculus

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)

 

\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)

 

\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)

 

\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)

 

\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)

 

\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

 

\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)

 

\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

 

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)

 

\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

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