Sadly, I’m coming to the end of another calculus course. I absolutely love teaching calculus, and I’m going to miss this particular group of kids a lot.

Today, after finishing a particularly effervescent inquiry into improper integrals, I asked my students to distill out what they would consider to be the “Central Theorem of Math Education.”

After some heavy processing, here’s what was said:

Lead us to the water, but don’t make us drink. Don’t even tell us that it’s the water we’re supposed to be looking at. For sure don’t hold our heads under and force us to drink the water level down until we’re not drowning anymore; that got really old.

The best part is that the investigation that precipitated that was actually fairly pedestrian.

# The Exposition:

We were toying with the idea of infinity, and we were playing the game of putting infinity places where it doesn’t belong. Like in denominators, exponents, and the like. Eventually I asked what would happen if you integrated to or from infinity.

Especially with something like:

[latex size=3]\int_0^{\infty} \! e^{-x} \, \mathrm{d} x[/latex]

Which, mind-blowingly, equals 1, btw, not infinity. The secret is in the concept of convergence/divergence, but THAT’S NOT THE SECRET.

The secret is getting students to yearn for the concept of convergence. This accidentally happened today, and it was awesome.

They were offended by the integral I just showed you. By all rights, that integral should yield infinite area.

They came up with other asymptotic-y equations and tried them. Most of them were members of this family:

[latex size=3]f(x)=\frac{1}{x^n}[/latex]

and

[latex size=3]\int_1^{\infty} \! \frac{1}{x^n} \, \mathrm{d} x[/latex]

The matherati among you are about to say, “Hey, this seems canned, you’re headed to a p-convergence-test lesson.”

Yeah, maybe you saw that coming, but I’m not that smart or clever.

All the kids did was follow a pattern. Which ones have answers, and which ones barf?

Turns out when the power is above 1, it gives an answer. When the power is 1 or below, it barfs. Who knew? I didn’t, because I hated Calculus II in college.

The kids even went so far as to discover that the ones that give answers (converge) have a patter to their answers: 1/(n-1)

In short:

- I did nothing
- The kids developed a pattern to test after I showed them something bothersome
- The kids confirmed their pattern with each other
- I ate a sandwich
- The students found a bonus pattern in the answers to their other pattern
- The students laughed at how the book stated the same idea (on page 504…)
- Students are primed for infinite series, and they don’t even know it