One of my students, K, and I were reviewing Taylor Series last Friday when she asked for a reminder why an infinite geometric series summed to for first term g and common ratio r when . I was glad she was dissatisfied with blind use of a formula and dove into a familiar (to me) derivation. In the end, she shook me free from my routine just as she made sure she didn’t fall into her own.
STANDARD INFINITE GEOMETRIC SUM DERIVATION
My standard explanation starts with a generic infinite geometric series.
We can reason this series converges iff (see Footnote 1 for an explanation). Assume this is true for (1). Notice the terms on the right keep multiplying by r.
The annoying part of summing any infinite series is the ellipsis (…). Any finite number of terms always has a finite sum, but that simply written, but vague ellipsis is logically difficult. In the geometric series case, we might be able to handle the ellipsis by aligning terms in a similar series. You can accomplish this by continuing the pattern on the right: multiplying both sides by r
This seems to make make the right side of (2) identical to the right side of (1) except for the leading g term of (1), but the ellipsis requires some careful treatment. Footnote 2 explains how the ellipses of (1) and (2) are identical. After that is established, subtracting (2) from (1), factoring, and rearranging some terms leads to the infinite geometric sum formula.
I despise giving any formula to any of my classes without at least exploring its genesis. I also allow my students to use any legitimate mathematics to solve problems so long as reasoning is justified.
In my experiences, about half of my students opt for a formulaic approach to infinite geometric sums while an equal number prefer the quick “multiply-by-r-and-subtract” approach used to derive the summation formula. For many, apparently, the dynamic manipulation is more meaningful than a static rule. It’s very cool to watch student preferences at play.
K understood the proof, and then asked a question I hadn’t thought to ask. Why did we have to multiply by r? Could multiplication by also determine the summation formula?
I had three nearly simultaneous thoughts followed quickly by a fourth. First, why hadn’t I ever thought to ask that? Second, geometric series for are absolutely convergent, so K’s suggestion should work. Third, while the formula would initially look different, absolute convergence guaranteed that whatever the “ formula” looked like, it had to be algebraically equivalent to the standard form. While I considered those conscious questions, my math subconscious quickly saw the easy resolution to K’s question and the equivalence from Thought #3.
Multiplying (1) by gives
and the ellipses of (1) and (3) partner perfectly (Footnote 2), so K subtracted, factored, and simplified to get the inevitable result.
That was cool, but this success meant that there were surely many more options.
Why stop at multiplying by r or ? Why not multiply both sides of (1) by a generic for any natural number N? That would give
where the ellipses of (1) and (4) are again identical by the method of Footnote 2. Subtracting (4) from (1) gives
There are two ways to proceed from (5). You could recognize the right side as a finite geometric sum with first term 1 and ratio r. Substituting that formula and dividing by would give the general result.
Alternatively, I could see students exploring , and discovering by hand or by CAS that is always a factor. I got the following TI-Nspire CAS result in about 10-15 seconds, clearly suggesting that
Math induction or a careful polynomial expansion of (6) would prove the pattern suggested by the CAS. From there, dividing both sides of (5) by gives the generic result.
In the end, K helped me see there wasn’t just my stock approach to an infinite geometric sum, but really an infinite number of parallel ways. Nice.
1) RESTRICTING r: Obviously an infinite geometric series diverges for because that would make as , and adding an infinitely large term (positive or negative) to any sum ruins any chance of finding a sum.
For , the sum converges iff (a rather boring series). If , you get a sum of an infinite number of some nonzero quantity, and that is always infinite, no matter how small or large the nonzero quantity.
The last case, , is more subtle. For , this terms of this series alternate between positive and negative g, making the partial sums of the series add to either g or 0, depending on whether you have summed an even or an odd number of terms. Since the partial sums alternate, the overall sum is divergent. Remember that series sums and limits are functions; without a single numeric output at a particular point, the function value at that point is considered to be non-existent.
2) NOT ALL INFINITIES ARE THE SAME: There are two ways to show two groups are the same size. The obvious way is to count the elements in each group and find out there is the same number of elements in each, but this works only if you have a finite group size. Alternatively, you could a) match every element in group 1 with a unique element from group 2, and b) match every element in group 2 with a unique element from group 1. It is important to do both steps here to show that there are no left-over, unpaired elements in either group.
So do the ellipses in (1) and (2) represent the same sets? As the ellipses represent sets with an infinite number of elements, the first comparison technique is irrelevant. For the second approach using pairing, we need to compare individual elements. For every element in the ellipsis of (1), obviously there is an “partner” in (2) as the multiplication of (1) by r visually shifts all of the terms of the series right one position, creating the necessary matches.
Students often are troubled by the second matching as it appears the ellipsis in (2) contains an “extra term” from the right shift. But, for every specific term you identify in (2), its identical twin exists in (1). In the weirdness of infinity, that “extra term” appears to have been absorbed without changing the “size” of the infinity.
Since there is a 1:1 mapping of all elements in the ellipses of (1) and (2), you can conclude they are identical, and their difference is zero.