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What are some good ways to motivate the material on infinite series that appears at the end of a typical American Calculus II course?

My students in this course are generally from biochemistry, computer science, economics, business, and physics (with a few humanities folks taking the course for fun) - not just math majors.

I have struggled some in the past to motivate the infinite series material to these students. For one, it doesn't fit with the rest of Calc II, which is on the integral. Over the years I have "converged" on telling them that the main point of the unit is Taylor series and that the rest of the material is there primarily so that we have the tools we need in order to understand Taylor series. Then I illustrate some of the many uses of Taylor series (mainly function approximation, at this level). This approach works better than anything I've come up with thus far with respect to getting my students to care about infinite series, but I feel a little like I'm selling the rest of the material short by subordinating it to Taylor series. Does anyone have other ways of motivating infinite series that they would like to share? (Again, only a small percentage of the students in my class are math majors.)

Background: The material in this unit typically consists of sequences, basic series (like geometric and telescoping ones), a slew of tests for convergence (e.g., integral test, ratio test, root test), an introduction to power series, Taylor and Maclaurin series, and maybe binomial series.

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I too usually like to start by talking about the local linear approximation, then a quadratic approximation, etc (that is, cover the material on Taylor polynomials), and then move on the Taylor series and talk about series as the tools to understand them, so that we can "write down" integrals that we cannot write down otherwise (e.g., $e^{x^2}$). – Arturo Magidin Nov 9 '10 at 4:08
I'm not too sure it's much of a motivation, but at least in chemistry and physics, people expand functions in series to truncate them (often severely) afterwards. For instance, there's the classical $\sin\;u\approx u$ that makes solving the differential equation for the pendulum (in the limit of small amplitudes) slightly more amenable. – J. M. Nov 9 '10 at 8:26
For chemistry, one encounters the ideal gas equation, $P\bar{V}=RT$ as a first approximation to the estimation of a gas's properties. However, this truncation is only useful for high temperatures ($T$) and low pressures ($P$), which is why we deal with so-called "virial expansions", $P=\frac{RT}{\bar{V}}\left(1+\frac{B}{\bar{V}}+\frac{C}{\bar{V}}+\dots\right)$, where $B,C,\dots$, are the second, third, ... virial coefficients. This accounts for the "nonideal" behavior of gases in practice. – J. M. Nov 9 '10 at 8:32
Computer science could be motivated by combinatorics, in turn motivated by Polya's enumeration theory, showing there is an equation for the set of all lists: L(x) = 1/(1-x). Although, admittedly, the theory itself requires a bit of pre-requisite material. So you couldn't squeeze it all into a single lecture. Perhaps a handout covering all the sciences could include a section on it. – Robin Hoode Nov 9 '10 at 13:51
@Mike I should probably add: While mentioning generating functions is probably good, giving an example of an actual data structure that has such a generating function is vastly better. Such computer science students taking Calc II will likely be well aware of data structures, but not of the fact that there are standardized ways of counting them, especially in the case where they haven't taken any courses that involve combinatorics. – Robin Hoode Nov 9 '10 at 22:44
up vote 15 down vote accepted

Thanks for all the responses so far. I thought I would summarize them for anyone else who might be interested in this question. Rather than continuing to update the summary in the original question, though, it seems better from an organizational standpoint for the summary to appear in a community wiki answer than I then accept so that it appears at the top of the list of answers.

  1. Others seem to agree that focusing on Taylor series is the right approach. See the answers and comments for justifications.

  2. Examples of the use of infinite series

    a. General: Zeno's paradoxes

    b. Physics: Using the first order Taylor approximation $\sin \theta \approx \theta$ in solving the pendulum differential equation

    c. Chemistry: Extending the ideal gas law to apply to high pressure and low temperature situations

    d. Economics: Calculating fiscal multipliers involves geometric series

    e. Computer science, 1: Several uses for generating functions (see examples by robinhoode and Raphael)

    f. Computer science, 2: Taylor series are involved in the error analysis of some numerical methods, such as Newton-Raphson and Simpson's rule.

    g. Mathematics, 1: Taylor series show that calculations involving functions like $e^x$ and $\sin x$ can all be computed using just addition, subtraction, multiplication, and division.

    h. Mathematics, 2: Power series, and Euler products in number theory in particular, as most people find number theory intrinsically interesting whether they have the background or not

    i. Mathematics, 3: Taylor series can be used to solve differential equations. (Often students will have seen a brief introduction to differential equations earlier in the course.)

    j. Mathematics, 4: There are infinite series expressions for interesting constants such as $\pi$ and $e$. Also, any nonterminating decimal representation of a real number is an infinite series.

    k. Mathematics, 5: Using Taylor polynomials to approximate integrands in definite integrals. (This fits well in a course like Calculus II that spends a lot of time on the integral.)

  3. Links to other resources

    a. Graphs of Taylor polynomials converging to a function (Brandon Carter's answer)

    b. Graphs of Taylor polynomials in the complex plane (Hans Lundmark's comment on Brandon's answer)

  4. Other

    a. Emphasize similarity of series and improper integrals

    b. Emphasize that convergence must be dealt with carefully. Use geometric series as a way to introduce convergence issues. Mention that even Euler was not always good about handling convergence.

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Actually, it looks like when you accept your own answer it doesn't move to the top of the answer list! I would appreciate it if people upvote this enough so that it does move to the top of the answer list. Note that this does not constitute begging for reputation points because this answer is community wiki. – Mike Spivey Dec 19 '10 at 23:35
I feel a bit depressed that people like Mike have to give disclaimers like his comment here... – J. M. Dec 20 '10 at 3:01
I don't see decimal expansions being represented as series in your list, after all that is all decimal expansions are, a short hand way for power series. The infinite decimals correspond to infinite series. – Arjang Jan 6 '11 at 6:06

While I somewhat struggle to relate to the problem (series was possibly the most interesting section of Calc II to me), I can share some insight on how my teacher was able to make them seem interesting. She used Mathematica demonstrations throughout every section of our calculus classes (I also had her for Calc III), and I found that the visualization of terms and partial sums helped to visually see the asymptotic behavior of each towards zero and the sum, respectively. Taylor and Maclaurin series were also very "unimportant" to me (in the sense that I did not at the time grasp their usefulness), until I was shown how the polynomials converged to the function. Some of her files are available on

Also, a key idea that has always stuck with me, is that we only know how to perform the basic arithmetic operations, and more complex functions like $e^x$, $\sin x$, etc. can all be computed using just addition, subtraction, multiplication, and division.

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Thanks, Brandon, and thanks for the links. I have also found that visualizing the Taylor polynomials converging to a function helps the students. As an aside, for some of my math majors infinite series is their favorite part of Calc II. I suspect that is because it is the deepest part of Calc II; it requires more rigorous thinking than the more computational material dealing with the integral. – Mike Spivey Nov 9 '10 at 4:26
@Mike Spivey: I don't know if your students know complex numbers yet, but the real fun begins when one sees how the Taylor polynomials approximate the function in the complex plane. (In case anyone is interested, I have some pictures on my web page: – Hans Lundmark Nov 9 '10 at 7:16
@Hans Lundmark: Unfortunately, they don't know complex numbers at this point. Thanks for the link to the cool pictures, though. – Mike Spivey Nov 9 '10 at 13:14
Just so you know, I "unaccepted" your answer yesterday. This is because I decided to reorganize my summary as a community wiki answer and mark that as my accepted answer, not because I was dissatisfied with your answer. – Mike Spivey Dec 20 '10 at 18:13
@Mike: No worries. I went ahead and +1'ed yours to try and move it to the top. – Brandon Carter Dec 20 '10 at 18:47

I think series do fit in with the integral quite nicely - the integral can be seen a "series on a continuous set of summands" while series are the discrete analog. They are both generalizations of finite everyday sums. The techniques used in the study of infinite series are similar to the ones used for improper integrals, and the integral convergence test is another reason why knowing integrals before studying series can be helpful.

As for why studying series is important, I think you should use motivating examples from the fields your students know, if possible. There is one basic "global" example I can think of - the paradoxes of Zeno.

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Thanks for the Zeno paradoxes idea. +1 for that alone. As far as "motivating examples from the fields your students know," part of the reason I'm asking this question is that I'm hoping to obtain some of those very examples from other Math SE users! And yes to the first paragraph; in fact, I do improper integrals right before series, so the students are already primed to see the similarities between the two. – Mike Spivey Nov 9 '10 at 5:34
My background is from computer science and I admit I cannot recall specific uses for series, except as a clever way to bound finite sums (e.g. the sum of an infinite geometric series is actually simpler than the sum of a finite one...) – Gadi A Nov 9 '10 at 13:58

I believe that power series were the main motivation for the formal study of infinite series; for instance, lots and lots of Euler's work consists of just multiplying out formal power series without any explicit concern for convergence. Perhaps demonstrating e.g. Euler products in number theory would show why we'd be interested in having a firm basis on which to rest such an argument.

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Thanks, but remember my audience mostly consists of non-math majors. I'm not sure how well a number theory justification would go over with them. – Mike Spivey Nov 9 '10 at 5:18
On second thought, this is useful. Specifically mentioning that great mathematicians haven't always used infinite series properly could help motivate the need for us to be particularly careful with them - and thus why we need things like the tests for convergence. – Mike Spivey Nov 9 '10 at 16:33
On top of that, I think we as mathematicians may develop an inferiority complex from hearing too many people ask whether individual concepts are relevant to their everyday lives. The sorts of people who take your calculus class sound like they're probably relatively curious people on average, and most people find number theory to be a relatively stimulating subject. You may be surprised by what people might be interested in. – Daniel McLaury Nov 10 '10 at 1:51
@user3926: That is a good point. In fact, I pulled Cantor's diagonalization argument on them a couple of weeks ago when we had five minutes left and I had finished saying what I wanted to say about that day's calculus topic. While I can't claim that they followed the whole argument I did have their rapt attention. :) – Mike Spivey Nov 10 '10 at 17:39

I want to give a more elaborate example for computer scientists.

Given a context free grammar, we can derive (in well-behaved cases) its structure function with a (often) simple calculation. This function's coefficients $a_n$ are the number of derivation trees for words of length $n$ (or, for unambiguous grammars, the number of words). With additional tricks, we can get a number of properties of the generated language.

Simple example: Let a grammar with the set of rules $S \rightarrow aS | bS | \varepsilon$. This directly translates to $S(z) = 2zS(z) + 1$. Solving this equation is simple; we obtain $S(z) = (1-2z)^{-1}$. The coefficients $[z^n]S(z) = 2^n$ are, of course, the number of different words of length $n$ this (unambiguous) grammar can generate.

This directly yields a convenient tool to check wether a grammar is unambiguous if we know the number of words (and we can solve the equation system and obtain the series expansion).

There are many further extensions and applications.

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Good question, and one that is timely for me as I'm teaching calc II next semester for the first time in several years.

An application of Taylor series that sometimes comes up in an intro to ODEs course is that of series solutions to differential equations. I usually don't spend more than a day or two on the topic, but I think the students appreciate seeing how a series representation of a solution can be found in cases that can't be solved by any of the other methods that they've learned. I especially like the example of Airy's equation $$ y'' - t \, y = 0 $$ because it's simple enough that students feel like there ought to be an explicit solution plus the series coefficients are fairly easy to express.

I'm not sure if this kind of application would be helpful or meaningful for calc II students, but maybe if they've seen some simple differential equations (separation of variables, for example) already?

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Thanks for your answer; I think I can use it. My students will generally have seen separation of variables earlier in the term, and so they will know what a differential equation is. – Mike Spivey Dec 17 '10 at 20:30

Some CS / computer engineering things:

  1. Newton-Raphson division is a clever way that you can do division with only using multiplication / addition. Telling people that that's how calculators do division might be interesting, and it is obviously relevant to ECE majors. (See the analysis of its error for how it relates to Taylor series.)
  2. You can use infinite sums to calculate pi, e, etc.
  3. More relevant to calc is the fact that numerical integration methods use e.g. Simpson's method, the error analysis of which is related to Taylor series.
  4. (Also, here Calc II is the same course people get introduced to linear algebra, and I wish someone had told me that PageRank was an eigenvector problem. So some unasked-for advice if it's the same where you are is to introduce eigenvectors this way)
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Riemann sums are finite, and sums over refined partitions do not correspond well to partial sums of series. So how are connecting Riemann sums of integrals to series? – Arturo Magidin Dec 17 '10 at 23:32
"It's been my experience that numerical analysis algorithms in general tend to use infinite sums" - I have to say not that much... the usual example of power series being used for evaluating special functions isn't entirely accurate, as power series are only useful near the expansion point. And, no, I don't remember the power method for dominant eigenpairs being connected in any way with infinite series. – J. M. Dec 18 '10 at 1:32
Regarding your parenthetical comment in 5: This is very much not the case in the U.S. Calc II is generally concerned with integration of real valued functions of real variable and its applications. There is no linear algebra at all (not even that of the real plane). – Arturo Magidin Dec 18 '10 at 2:12
@J.M.: I should've been more clear that my experience was the result of taking "numerical analysis for computer scientists," which, at least at my uni, is applications of Taylor's theorem. I mentioned it as OP was interested in applications for non-math-majors; I didn't intend to proclaim what "real" numerical analysis studies, as I know nothing about that subject. – Xodarap Dec 19 '10 at 2:40

I agree with the sentiment that power series is the best way to motivate infinite series.

There is a natural progression from the elementary operations of addition, subtraction, and multiplication; to the concept of a polynomial. A polynomial is the easiest way to combine the concept of a variable with addition, subtraction, and multiplication.

In the same way, power series are the most natural way to combine the concepts of a polynomial with an infinite limit. Since any (good) calculus course deals with derivatives as limits, the students at this level already should know about limits; and it's not so hard to combine that with polynomials in the obvious way.

I think when students first encounter infinite series, it's best to caution them strongly that convergence is something that has to be dealt with in a very careful, rigorous way; and then to introduce the geometric series and use it as an example to discuss the basic ideas of convergence\divergence.

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Thanks for your answer. – Mike Spivey Jan 5 '11 at 3:59

An example from economics:

Suppose the government gives you a 100 dollar tax rebate. You will save, say, 10, and spend the other 90. So at first glance, it seems that the 100 dollar rebate increased GDP by 90 dollars.

But the 90 you spent will not just disappear; it will go to other people, who will in turn save 10% of it, and spend the other 90%. The 90% the second round spent will go to a third round of people who spend 90% of that, which goes to a fourth round of people... well, you see where I'm going with this.

$\begin{align} \Delta GDP & = 90+81+72.9+\dots \\ & = 100(0.9^1 + 0.9^2 + 0.9^3 \dots) \\ & = 100\left(\lim_{n\to\infty}\sum_{i=1}^{n}0.9^i\right) \\ & = 100 \left(\frac{1}{1-0.9}\right) \\ & = 100 (10) \\ & = 1000 \end{align}$

So we can see that the fictional stimulus package of a hundred dollars led to an increase in GDP of a thousand dollars..

If you are interested, the "90%" is what's called the marginal propensity to consume and the sum of the series (10 in this example) is the multiplier. At my school, this is taught in econ 101 without giving the basis (people are told to just memorize $\frac{1}{1-MPC}$) so some of your students may be interested to learn why this formula is correct.

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Thanks! It's nice to have an example from another field. – Mike Spivey Jan 5 '11 at 3:56

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