# Students in the Comparison Test

Background: I have two students in Calculus II.* A boy, M, and a girl, G, who have been really good friends for a very long time. They’re very comfortable with each other.

Today we hit the Direct Comparison Test.** (And the Limit Comparison Test, too.) The quick-and-dirty summary of the DCT says that if you’re trying to determine if a series converges or not, compare it to a different series. If your original series is smaller than your new series and the new series converges, then so does the original. If your original series is bigger than your new series and the new series diverges, then so does the original.

The standard statement of the theorem says it something like this: If $\{a_n\}$ and $\{b_n\}$ are sequences and $a_n \leq b_n$ for all n, then …. . But I have a goal this year of making less complicated and more memorable statements. So…the smaller series became G and a the bigger series became M. Because in my classroom, M is bigger than G.

Is it anything special? Eh, probably not, but it made all three of us smile.

Of course, I’m writing this so that I’ll remember it for next year, and the two students taking this class next year are about the same size, so I’m not sure what I’ll do then.

*Calculus is AP Calc AB. Calc II is roughly BC, but it’s for students who have already taken AB, so it’s everything in BC that’s not in AB. Which is a ton of integration methods, polar, vectors, and series. We covered the first three last semester. I have no idea what we’re going to do between series and AP review, because there’s going to be a ton of time. Graph theory? 3d graphing? Combinatorics?

**After going over this today, I’m wondering: why do we need the direct comparison test if we have the limit comparison test? The limit comparison test lets you get away with murder and still come up with an answer! Is it because we use the direct comparison test in the proof for the limit comparison test? Or is there a series where the direct comparison test works when the limit comparison test fails?