Motivating infinite series

Question

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.

Answer 1

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.

Answer 2

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 demonstrations.wolfram.com:

http://demonstrations.wolfram.com/SeriesAFewExamples/ http://demonstrations.wolfram.com/SeriesStepsOnANumberLine/ http://demonstrations.wolfram.com/GraphsOfTaylorPolynomials/

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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?

Answer 7

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)

Answer 8

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.

Answer 9

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.

Answer 10

From the point of view of a student that is struggling with the abstract concepts of maths, I found that using Zeno's Paradox was an interesting approach to infinite series.