My Favorite One-Liners: Part 5

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.

Every once in a while, students encounter a step that seems far from obvious. To give one example, to evaluate the series

\displaystyle \sum_{n=1}^{100} \frac{1}{n^2+n},

the natural first step is to rewrite this as

\displaystyle \sum_{n=1}^{100} \left(\frac{1}{n} - \frac{1}{n+1} \right)

and then use the principle of telescoping series. However, students may wonder how they were ever supposed to think of the first step for themselves.

Students often give skeptical, quizzical, and/or frustrated looks about this non-intuitive next step… they’re thinking, “How would I ever have thought to do that on my own?” To allay these concerns, I explain that this step comes from the patented Bag of Tricks. Socrates gave the Bag of Tricks to Plato, Plato gave it to Aristotle, it passed down the generations, my teacher taught the Bag of Tricks to me, and I teach it to my students.

Sadly, there aren’t any videos of Greek philosophers teaching, so I’ll have to settle for this:

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