Celebrating Pi day with a fun calculus paper from Susan Jane Colley

About a week ago I saw a great tweet from Dave Richeson about some journal articles that were being made freely available for the month of March:

The paper from the The College Mathematics Journal by Susan Jane Colley caught my attention for being both an really neat result and being explained at a level a student taking calculus could understand.

So, this morning to celebrate Pi day I decided to use the paper to talk a bit of calculus with my son. Pulling all of the different ideas together was challenging for him, so we went slow but still made it through the main points in about 30 min.

We took a quick look at the paper and then started digging into the math by looking at the famous alternating series for \pi / 4.

I should say for clarification that I forgot to look up Susan Jane Colley’s current position before we started the project and wasn’t sure if she was still at Oberlin or had moved to a different university since the paper was published in 2003. But to be clear, she is the Andrew & Pauline Delaney Professor of Mathematics at Oberlin.

Next we dove in to the connection between the alternating series and \pi. I thought I’d try to introduce the connection in a sneaky way, but it was sort of a dead end. Eventually, though, he thought about arctangent.

At the end of the last video the formula for an infinite geometric series came up, but that formula wasn’t quite at the top of his head. So we took a little detour to re-derive that formula. Once we had that formula we could see that the alternating series we were looking at converged to \pi/4:

Now we looked at the main result of the paper – a different series for \pi that converges really fast.

Here we look briefly at the formula for this series (sorry for the reading typos by me – trying to read the paper and stay behind the tripod and not cast a shadow was hard . . . . )

Finally, we went to Mathematica to evaluate the integral and look at the speed of convergence of the two series we’ve been studying:

I think that Colley’s paper is absolutely terrific and a great resource to use to show calculus students some advanced math. It is an extra terrific resource to use on Pi day 🙂