How Much Mathematics Should A Student Memorize? Part 4, Geometric Series

In teaching mathematics for many years, one of the things I emphasized over and over again was that students should memorize the absolute minimum necessary, and then I did my best to make explicit what this absolute minimum is. It is better, I explained, to spend time solving problems, discussing applications, “reading around the subject,” and so on, rather than waste time on sterile memorization.

And certainly the best way to memorize something in mathematics is to use it repeatedly, meaningfully. After teaching a subject repeatedly, one naturally memorizes things.

Having said all this, I have to admit that sometimes I lied in order to drive home this point. That is, I sometimes pretended that I did not remember certain things, in order to work them out on the spot, in an attempt to illustrate the point that it was not necessary to have memorized them in the first place. Sorry!

But there is one formula that I have real problems remembering, and so I really do have to work it out from scratch every time I want to use it (looking it up is too much of a hassle, and it is really fast to work out, as I’ll show you in  a moment). It’s the formula for the sum of a finite geometric series.

The trick to work out the formula really is easy to remember, and it works for both finite and infinite geometric series. And this includes repeating decimals, which are a kind of infinite geometric series.

First, to get the formula for the sum of an infinite geometric series, call the sum $S$:

$S = 1 + r + r^2 + r^3 + \ldots $

Now multiply both sides of the previous equation by $r$:

$rS = r + r^2 + r^3 + r^4 + \ldots $

Subtract the second equation from the first, noticing that all of the terms except one cancel on the right side:

$S – rS = 1$

which means that

$S = \dfrac{1}{1 – r}$

Note that this trick is just a memory aid, and the verification that the formula really is valid (for $-1 < r < 1$) requires a certain amount of formal justification. But the nice thing is that the very same trick works for finite geometric series:

$S = 1 + r + r^2 + r^3 + \ldots + r^n$

Now multiply both sides of the previous equation by $r$:

$rS = r + r^2 + r^3 + r^4 + \ldots + r^n + r^{n+1}$

Subtract the second equation from the first, noticing again the nice cancellation on the right side:

$S – rS = 1 – r^{n+1}$

Solving for $S$ gives the desired formula:

$S(1 – r) = 1 – r^{n+1}$

$S = \dfrac{1 – r^{n+1}}{1 – r}$

Full disclosure: I remember the structure of this formula, but I don’t have confidence in the exponent of $r$ in the numerator. Is it $n$ or is it ($n$ + 1)? I’m never sure, so I have to work it out from scratch each time. I speculate that the reason for this is that I’ve seen the formula too many times in too many books in more than one form. Some books stop at $r^{n-1}$ instead of $r^n$ in the setup, and so they end up with a different formula. I’m too lazy to try to sort this through (but really, how hard is it? Not very …), so I just work it out every time. Thankfully, it only takes seconds, which justifies (don’t you agree?) my not bothering to carefully remember the formula.

Delightfully, the same trick allows one to easily convert a repeating decimal to a fraction. For example, consider the repeating decimal $0.34343434343434$ … . You can see that this is just a geometric series, right? Then the same trick should work. Call the decimal $S$, then multiply both sides of the resulting equation by $100$ (why $100$?), and then subtract and solve for $S$:

$S = 0.34343434 \ldots$

$100S = 34.34343434 \ldots$

$100S – S = 34$

$99S = 34$

$S = \dfrac{34}{99}$


ps. Back to my exhortation to students to “memorize the absolute minimum.” Sometimes in class I would solve problems in class in two ways; one relying on the memorization of a formula (usually a standard formula highlighted in the course textbook), and the other without using the formula. After presenting both solutions, I would poll the class and ask which solution was preferred, and almost always the majority of the class preferred the solution that relied on the memorization of a formula. This is part of human nature, I believe: thinking is hard. It is easier for many people to memorize formulas (which are no doubt forgotten soon after the final exam) than to truly understand the subject. But this is not the fault of our students, but rather the fault of the structure of our school system, which relies on high-stakes testing (much of it which emphasized technique over deep understanding), heavily penalizes mistakes, and rushes students through courses at such a rapid pace that very few have much chance of truly understanding a course well before it ends. (I’ve discussed aspects of this problem a number of times, including yesterday.) A lucky few (primarily teachers!) have the opportunity to go over the material again some day, and then their understanding grows.

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Earlier posts in this series:

How Much Mathematics Should a Student Memorize? Part 3, The Graphs of Power Functions

How Much Mathematics Should a Student Memorize? Part 2, Integral Calculus

How Much Mathematics Should a Student Memorize? (Part 1, Trigonometric Identities)

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 22 January 2021.)