The other day, while writing a quiz for the applied calculus 2 course that I lecture for, I became bored, as instructors often do when teaching a skill that they now find trivial on the other side of many tedious hours of practice. I decided to write a fraction to troll my students. So I asked the very natural question,
What is $0.\overline{67}$ as a fraction?
Recall the formula for evaluating geometric series: $$\sum_{n = 0}^\infty a_0 r^n = \frac{a_0}{1 - r},$$ We can use this to write our decimal as a fraction. Because we are in base ten, we can write our decimal as a power series like so: $$\begin{align*} 0.6767\ldots &= \sum_{n = 1}^\infty \frac{67}{100^n} \end{align*}$$ If this equality doesn't make sense to you, write out the first two or three terms to convince yourself that this is true. Now it is a matter of making our series look more like the left-hand side of the formula to identify $a_0$ and $r$. We start our series at $n = 0$: $$\begin{align*} \sum_{n = 0}^\infty \frac{67}{100^{n + 1}} &= \frac{67}{100} \sum_{n = 0}^\infty \left( \frac{1}{100} \right)^n \end{align*}$$ So since $a_0 = 67/100$ and $r = 1/100$, we plug this into our geometric series formula to get $$0.\overline{67} = \frac{67}{99}$$ Thanks to my honors calculus 2 professor Pablo Bert for teaching me this neat trick, and thanks to my applied calculus 2 students for putting up with my shenanigans. What a full-circle moment that helps me have a little fun and hopefully give a chuckle during the stress of the quiz. You too should evaluate the integral $$\int_0^\infty \frac{67x}{11 e^{3x}} dx.$$