Calculus II Power Series Projects

In my Calculus II course, I divide the class up into small groups. Each group works on one of the projects below (assigned as best as possible to accomodate everyone’s preferences). Then, each group presents their work in class in our Power Series Miniconference! Here are brief descriptions of each project along with a link to the project. The first three are TRIUMPHS Primary Source Projects. The rest are provided as read-only links to the Overleaf document in which I created them.

  • Euler’s Sum of the Reciprocals of the Squares by Kenneth M Monks. The divergence of the harmonic series, the fact that $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots = \infty,$$ has been known since the 1400s. A natural followup is to study the sum of reciprocals of squares: $$1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\cdots.$$ The standard Integral Test for power series convergence can be used to show that their total converges to a finite value, but what exactly is that value? In this project, we trace through Euler’s original calculation of that mysterious summation. Shockingly, the key object used is the power series of the function $\sin(x)/x$.
  • How to Calculate $\pi$: Machin’s Inverse Tangents by Dominic Klyve. Here we explore how John Machin, an 18th century English astronomer, used Leibniz’s power series formula for arctangent to calculate $\pi$. Specifically, in 1706, he became the first person ever to calculate it to 100 decimal places!
  • Euler’s Rediscovery of $e$ by Dave Ruch. This project shows Euler’s use of infinitely large numbers, infinitely small numbers, and the binomial series to construct the number $e$. Note that Dave’s original is intended for an upper-level analysis class. This is a modified version appropriate for use in a second-semester calculus class.
  • Infinite Series in Probability by Kenneth M Monks. This project guides a student through the introductory ideas of probability and shows how one can use power series to compute expected values.
  • Bhaskar’s Approximation to Sine by Kenneth M Monks. Hundreds of years before the development of power series, Bhaskar (7th century India) constructed a rational function that approximates sine with astonishing accuracy. Specifically, he approximated sine for real numbers $x\in[0,\pi]$ as $$\sin(x)\approx \frac{16x(\pi-x)}{5\pi^2-4x(\pi-x)} .$$ This project guides the reader through one possible process by which he could have come up with the formula.
  • Partition Counting via Generating Functions by Kenneth M Monks. A partition of a natural number $n$ is an expression of $n$ as a sum of nonzero natural numbers, perhaps including itself. For example, the number $n=5$ can be partitioned in seven ways: $$5=4+1=3+2=3+1+1=2+2+1=2+1+1+1=1+1+1+1+1.$$ For larger $n$, listing out partitions becomes completely intractible. In this project, we demonstrate how one can count partitions (even for quite large $n$) by taking products of geometric series.
  • Proof of the Irrationality of $e$ by Kenneth M Monks. This project uses both the exponential and the geometric series to prove that $e$ cannot be written as a ratio of integers. We follow an argument similar to Joseph Fourier’s (18th century France) famous proof by contradiction.

Differential Equations with Linear Algebra OER

My MAT266 differential equations with linear algebra course runs via eight Overleaf projects. In these projects, students find notes, worked out examples, exercises, and comment boxes in which to review each other’s work and ask questions. The course is self-contained in these documents and does not require a textbook! Feel free to contact me if you are teaching differential equations and would like to use these projects; I am happy to discuss any implementation details you might be wondering about.

LaTeX/Overleaf Quick Start Guide

Since I require LaTeX and Overleaf in my Calculus III and Differential Equations courses, I provide a quick start guide as the first assignment of the semester.