Dentistry and Math

Back in the early '70s, I used to hang out some evenings (once a week maybe?) at the UH math department, where someone would give a talk and then we would go to "Chico's" at the foot of St. Louis Heights (where City Mill is today). Most of the guys would drink beer but I'd have a 7-up or similar (I was under age).

These were interesting characters. One of them, a PhD student, said on one evening that he didn't believe the fundamental theorem of calculus. I don't know how you can be working on a Ph.D in mathematics and not believe the fundamental theorem of calculus, but there you are.

One evening, a professor (it might have been Dr. Wallen, department chair) told us about his dentist. (I've told this story 2-3 times on this vacation, and as the lovely Carol points out, I laughed out loud each time, so I decided I should write it down.) So he was at the dentist one day, and the dentist asked (you know how they do, when your mouth is full of whatever and you can't answer), "So what do you do?"

"Mf-fa-muh-fix," he replied.

The dentist made some enthusiastic noises, and the professor thought, yeah right, and endured the rest of his dental treatment. But as he was leaving, the dentist said, "Wait, when can I come over?"

Come over?

"Yeah, come over and do math. I have some papers..."

So at the appointed hour some days later, the prof opens the door to find his dentist with a sheaf of papers and a bottle of vodka. (This is the part where I always laugh. Dentistry, vodka, math -- what a great combo!) So they drink and the prof looks over the dentist's papers.

Relating this story to us, the prof said, "This guy was powerful." He had worked out a conjecture regarding the infinite series Σ_n 1/n^x; he wasn't sure, but it seemed to him that if x was bigger than 1, then the series would converge, otherwise not.

It turned out that this was correct; if x=2, the sum converges to π²/6; if x=1, it diverges; if x=–1, it converges, but I forget what it converges to. I thought this was called "Cauchy's theorem," but a quick search (it has been about 35 years) doesn't connect that name to that theorem.

The most recent occasion for telling this story was when talking with my nephew (currently taking high school calculus) about some of my experiences "when I was your age." It was fun re-living those experiences, and also helping him review some of his math and giving him a preview of the fundamental theorem of calculus (which I actually do believe, ftr).

Oops! November 2014 update

OK, that part about –1 is completely wrong. What I meant was, the series 1/1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6… which is not Σ_n 1/n^–1; it's actually Σ_n 1/(-n)¹