My Favorite One-Liners: Part 29

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. Today’s quip is one that I’ll use when I need my students to remember something from a previous course — especially when it’s a difficult concept from a previous course — that somebody else taught them in a previous semester.

For example, in my probability class, I’ll introduce the Poisson distribution

P(X = k) = e^{-\mu} \displaystyle \frac{\mu^k}{k!},

where \mu > 0 and the permissible values of k are non-negative integers.

In particular, since these are probabilities and one and only one of these values can be taken, this means that

\displaystyle \sum_{k=0}^\infty e^{-\mu} \frac{\mu^k}{k!} = 1.

At this point, I want students to remember that they’ve actually seen this before, so I replace \mu by x and then multiply both sides by e^x:

\displaystyle \sum_{k=0}^\infty \frac{x^k}{k!} = e^x.

Of course, this is the Taylor series expansion for e^x. However, my experience is that most students have decidedly mixed feelings about Taylor series; often, it’s the last thing that they learn in Calculus II, which means it’s the first thing that they forget when the semester is over. Also, most students have a really hard time with Taylor series when they first learn about them.

So here’s my one-liner that I’ll say at this point: “Does this bring back any bad memories for anyone? Perhaps like an old Spice Girls song?” And this never fails to get an understanding laugh before I remind them about Taylor series.


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