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Infinite Series is an unusual calculus topic but series can be very useful for computation and problem solving, especially when it comes to integration and differential equations. However, in the realm of infinity, unusual things start to happen. So more care is required. Also, when you start working with infinite series, the question of convergence becomes important. This page covers the introduction to infinite series and contains a lot of basic information to get you started.
Getting Started |
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To get started, let's watch some videos. Because of the complexity of this topic, it is best to watch all of these introduction videos. There is some repetition but the information comes with different viewpoints and it will help you to assimiliate the information if you watch all of them.
This first video is a very basic introduction to series. If you have some experience with series and know how to expand and use the terms, you can skip this video.
In this video he introduces sequences, series and sigma notation. He does a pretty good job explaining the basics required for understanding series.
About 3 minutes into the video, he says something that is correct in this context but not in general. He says that you can add numbers in any order and get the same answer. This is true for a finite list of numbers, like he is doing here, but it is NOT always true if you have an infinite list of numbers. You will probably run across this case in your textbook while you are studying this topic.
video by Khan Academy
This second video contains a quick introduction to series and uses the geometric series to show some examples.
video by PatrickJMT
This next video is the most important introduction video. If you watch only one, this is the one to watch. This video gives a great introduction and overview of many of the series tests.
video by Dr Chris Tisdell
This video contains important theory that will help you get a good grasp of series. He uses improper integrals to explain the theory of series and introduce the ideas of convergence and divergence. The presenter is clear, communicates well and is interesting to listen to.
video by MIT OCW
This last introduction video is really good to watch and contains a visual example of what he calls a visceral example of infinity. It will help you (a lot) to get a feel for series. The presenter is the same person as the previous video (this is actually a continuation of the previous video) and is quite interesting. This video is recommended but not required for the rest of your work with infinite series.
video by MIT OCW
Okay, now that you have a grounding on the basics, let's get down to some details. You can go through these next two panels now. We suggest you scan the first panel (big picture) and read the second panel (study techniques) thoroughly.
The study of Infinite Series is perhaps the most difficult material of all of calculus. It will help you to have a big picture. The study of series can be divided into 3 parts.
Part 1 - Introduction |
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Section 1: Sequences
This is the only section on sequences and they are introduced as an underlying structure of series.
Part 2 - Special Series and Series Tests |
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These topics will probably appear in a different order in your textbook. They are grouped here by type. As you learn the series in this section, notice that we are looking at series that contain only numbers and indices. So, if the series converges, it will converge to a number. Sometimes we know this number or can find it. More often, though, we don't know what the number is. We are interested only in whether the series converges or diverges.
Hint: Get to know the Ratio Test very well. You will be using it a lot in Part 3.
Section 2: Series and Convergence
This section usually introduces the series and their relationship to sequences, as well as the idea of convergence.
Section 3: p-Series
This section begins the discussion of specific types of series and under what conditions they converge and diverge. Knowing these is important since you will use them later to determine convergence or divergence of series that are not in a special form.
Section 4: Geometric Series
This is an extremely important series that is used extensively when working with infinite series, Taylor Series and Power Series.
Section 5: Telescoping Series
This is a special series the doesn't show up much but you can learn a lot about how series work when you study this type of series. Also, if you recognize that your series is a telescoping series, it is easy to determine what it converges to.
Section 6: Alternating Series
This is a unique type of series and begins the the discussion of series tests.
Section 7: Ratio Test
Of all the tests, this is, by far, the most important and most used. It is critical that you understand this series and how to use it. Also, my experience has shown that often textbooks will combine this test in the same section with the Root Test, which is the least used test.
Section 8: Series Comparisons
This section discusses two important comparison tests, the Limit Comparison Test and the Direct Comparison Test. These are very important tests but require a bit of study to get your head around them (especially the Direct Comparison Test).
Section 9: Integral Test
This is a fairly useful test but, as you know from your study of integration, it is not always possible to integrate functions. So I usually try another test before using this one.
Section 10: Root Test
Finally, this test has limited usefulness but can come in handy under certain conditions.
Part 3 - Applications of Series |
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In this third part, notice that we are now working with series that contain a variable ( usually x ). So we will end up with a function, not a number, if we know what the series converges to. This is an important distinction that may not be obvious when first learning these sections.
Section 11: Taylor Series
Sometimes this material is broken into two additional subsections, Taylor Polynomials (finite series) and Taylor Series (infinite series).
Section 12: Power Series
This is a great section for representing functions as Power Series (Geometric Series).
In both Sections 11 and 12, you will probably find some discussion about approximating functions and series and the resulting error. This material looks harder than it is. So read the material on this site and see if it helps you.
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.
If you use these study techniques, 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?
Infinite Series Properties |
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When working with infinite series, it is not always possible to determine what a series converges to, if it converges. It is enough to know whether it converges or diverges. Here are some important properties of series that will help you as you work with them.
For the following properties, we will assume that we have two convergent series \(\sum{a_n}\), which converges to \(A\) and \(\sum{b_n}\), which converges to \(B\). We also have a real number \(c\).
1. The series \(\sum{[c\cdot a_n]}\) converges and \(\sum{[c \cdot a_n]} = c\sum{a_n} = cA\). |
2. The series \(\sum{[a_n \pm b_n]}\) converges and \(\sum{[a_n \pm b_n]} = \) \(\sum{a_n} \pm \sum{b_n} =\) \(A \pm B\) |
3. Adding or subtracting a finite number of real numbers to a series does not change whether or not it converges or diverges. |
4. If we have a divergent series \(\sum{d_n}\), \(\sum{a_n} + \sum{d_n}\) diverges. [ link to quick proof by contradiction ] |
Infinite Series Reference |
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These remaining panels below require more knowledge of various series tests before they make any sense. We suggest that you scan this material so that you know what is here and then come back as you need to.
How do you know which test to use to determine convergence or divergence of an infinite series? Knowing which test to use is a combination of practice and guessing. However, keep in mind that there may be several tests that will work but there is often a best test that is more efficient and quicker than using other tests.
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.
video by PatrickJMT
With most series, you will be asked only to determine if the series converges or diverges. However, you will encounter some series where you are asked to determine the value that the series converges to. So, how do you know if you can determine that value or not? There are two types of series where it is possible to determine the convergence value.
Geometric Series - If you can get your series into the form of a geometric series, then you can determine the convergence value and the convergence interval. This is the technique used for determining power series.
Telescoping Series - With telescoping series, your series must already be a telescoping series in order to determine the convergence value. This technique usually involves writing out several terms to determine if any terms cancel. Then taking the limit of the remaining terms.
NOT The Integral Test - One thing that confuses some people while they are learning infinite series techniques is what to do with the result of the integral test. If you use the integral test and you get a finite value, the significance of that value is only that it is finite. The value is NOT what the series converges to.
IMPORTANT!
When determining only convergence or divergence of an infinite series, the starting value of n does not matter, i.e. whether your series starts with zero, one or any other number, the convergence or divergence is not changed. HOWEVER, when you talk about what a series converges to, you must take into account the starting value. So watch that carefully. Some teachers will put a series on the homework or exam that is familiar but then they will change the starting value just to see if you are paying attention.
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
this 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
the practice list 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.
Infinite Series Table |
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Here is a table of series and tests. The table is arranged in order of use, as described above in choosing a test. If you are already familiar with the tests, there is enough detail in this table to allow you to use it rather than flipping through the pages in your textbook. You may download a pdf version of this table. Make as many copies as you want and share it with your friends and classmates. For teachers, feel free to make copies for your students and use it in your class. All we ask (from both students and teachers) is that you keep the 17calculus.com information at the bottom of the page.
Group 1 - First Test You Should Always Try | |
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Divergence (nth-Term) Test | |
convergence: cannot be used | |
divergence: \(\displaystyle{ \lim_{n \to \infty}{ a_n \neq 0} }\) | |
Group 2 - Special Series | |
p-Series | |
convergence: \( p > 1 \) | |
divergence: \( p \leq 1 \) | |
Geometric Series | |
convergence: \( |r| < 1 \) | |
divergence: \( |r| \geq 1 \) | |
sum: \(\displaystyle{ S = \frac{a}{1-r}; ~~ |r| > 0 }\) | |
Alternating Series | |
convergence: 1. \( \lim_{n \to \infty}{a_n=0}\) and 2. \(0 < a_{n+1} \leq a_n \) | |
divergence: cannot be used | |
condition 1 is the divergence test; remainder: \( |R_N| \leq a_{N+1} \) | |
Telescoping Series | |
convergence: \(\displaystyle{ \lim_{n \to \infty}{b_n} = L }\) | |
divergence: cannot be used | |
\(L\) is finite; sum: \(S=b_1 - L\) | |
Group 3 - Core Series Tests (in order of use) | |
Ratio Test | |
convergence: \(\displaystyle{ \lim_{n \to \infty}{\left| \frac{a_{n+1}}{a_n} \right| } < 1 }\) | |
divergence: \(\displaystyle{ \lim_{n \to \infty}{\left| \frac{a_{n+1}}{a_n} \right| } > 1 }\) | |
test is inconclusive if \(\displaystyle{ \lim_{n \to \infty}{\left| \frac{a_{n+1}}{a_n} \right| } = 1 }\) | |
Limit Comparison Test | |
convergence: \(\displaystyle{ 0 \leq \lim_{n \to \infty}{ \frac{a_n}{t_n} } < \infty }\) and \(\displaystyle{ \sum_{n=1}^{\infty}{t_n} }\) converges | |
divergence: \(\displaystyle{ \lim_{n \to \infty}{ \frac{a_n}{t_n} } > 0 }\) and \(\displaystyle{ \sum_{n=1}^{\infty}{t_n} }\) diverges | |
\(\displaystyle{ \sum_{n=1}^{\infty}{t_n} }\) is the test series, \(a_n > 0\) and \(t_n > 0\) | |
Direct Comparison Test | |
convergence: \( 0 < a_n \leq t_n \) and \(\displaystyle{ \sum_{n=1}^{\infty}{t_n} }\) converges | |
divergence: \( 0 < t_n \leq a_n \) and \(\displaystyle{ \sum_{n=1}^{\infty}{t_n} }\) diverges | |
test series: \(\displaystyle{ \sum_{n=1}^{\infty}{t_n} }\); \(a_n, t_n > 0\) | |
Integral Test | |
convergence: \(\displaystyle{ \int_{k}^{\infty}{f(x)~dx} }\) converges | |
divergence: \(\displaystyle{ \int_{k}^{\infty}{f(x)~dx} }\) diverges | |
\(f(x)\) is continuous, positive and decreasing for \(x > k\); \(a_n = f(n) \geq 0 \) | |
Root Test | |
convergence: \(\displaystyle{ \lim_{n \to \infty}{ \sqrt[n]{|a_n|} } < 1 }\) | |
divergence: \(\displaystyle{ \lim_{n \to \infty}{ \sqrt[n]{|a_n|} } > 1 }\) | |
inconclusive: \(\displaystyle{ \lim_{n \to \infty}{ \sqrt[n]{|a_n|} } = 1 }\) |