Motivating infinite series

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.

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.

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.