Our introductory mathematics courses are filled with finished products—not only are our proofs and procedures refined, but so too are the formulae themselves.
This post gives one example of such a perfect formula along with the question: what might we learn from starting with a much weaker version of it?
The formula for the sum of a geometric series with first term a and common ratio r, is well known. Perhaps you can write it down. The formula is perfect—it provides answers efficiently and the proof is easy to understand.
Now pretend that you don’t know it. Instead you are familiar with just one infinite sum:
Perhaps you discovered it on your own and don’t quite know what it means (suppose you haven’t yet learned about convergence). Nevertheless, let’s see how far you can get with it—an imaginary contest, if you will, between the use of the standard perfect formula and your smaller formula (the one above).
The most famous geometric series is, of course:
Your formula, with x=2, wins—no contest. (I’m deliberately not providing answers). How about:
Well there are two options for how to adapt your formula. Either add 1 to the answer from eg.1, or else multiply every term from eg. 1, (and hence also the answer).
After experimenting on your own examples, you may find that multiplying every term (and hence the answer) by a constant is very useful!
Now for an alternating series:
How about using x=–2 and then multiplying the terms (and hence the answer) by –1? Are there other strategies? Your formula still seems to have the advantage. Let’s try finite sums. This one has only 5 terms:
Piece of cake! Use the difference of two infinite sums.
Now for another finite sum and something really cool:
Do we dare use the difference of two infinite series again? Let’s try:
Hmm. Using x=1/2 gives the right answer but some of the steps don’t seem right!
Here’s another option:
Easy—we’ve seen the sum in the brackets before.
The first attempt was still useful. It may help us to understand this one:
Here set x=3/2 and the answer comes out right away.
Can you find an example where the standard perfect formula is really better? (Hint: I can’t). The restriction required for convergence (x>1) seems very natural. The process of adapting the formula by shifting it in various ways is familiar to a mathematician and it is fun; and now you have your own questions to answer: what question might lead a student to discover this simpler formula in the first place? Do you have a direct simple proof? Try it!