Primer on Infinite Series

What does it mean to add things to­gether for­ever? While many get in­tro­duced to in­finite se­ries in Calcu­lus class, it seems that ev­ery­one skips ahead to mem­o­riz­ing ter­minol­ogy with­out first mo­ti­vat­ing the sub­ject, or con­sid­er­ing that it’s sorta weird. In re­al­ity, the long his­tory of in­finite se­ries would sug­gest that the sub­ject is sub­tle and strange. But by steam­ing ahead, mem­o­riz­ing the vo­cab­u­lary, ex­am­ples, and rules for con­ver­gence, learn­ers miss a chance to re­ally think about what these in­finite sums re­ally are or could be. They end up be­ing able to more or less ac­cu­rately par­rot those rules, but if you speak with them a bit or over­hear a con­ver­sa­tion with oth­ers about the topic, you re­al­ize their un­der­stand­ing is a hol­low shell. This ar­ti­cle is a look at in­finite se­ries, adding things to­gether for­ever, from more or less first prin­ci­ples.

Ex­am­ple One: Zeno Crosses the Room

One of the ori­gin sto­ries for in­finite se­ries is at­tributed to the Greek Philoso­pher Zeno. It’s a good place to start mo­ti­vat­ing why you end up deal­ing with in­finite se­ries and also a place from which you can be­gin to ap­pre­ci­ate the need for a way of think­ing about in­finity that doesn’t lead to pure non­sense. Although he had many para­doxes, they all boil down to cer­tain weird­nesses that come up when you imag­ine space or time be­ing in­finitely di­visi­ble. Things that are ob­vi­ous and sim­ple from the point of view of ev­ery­day ex­pe­rience are con­tra­dicted by their anal­y­sis from the point of view of in­finite di­visi­bil­ity. Con­sider the fol­low­ing sce­nario.

You want to cross the room.

That is you wish to cover one room’s worth of dis­tance. Easy enough right? You don’t even need to stretch. But…(Zeno’s voice) to cross the room, first you need to get halfway there. Be­fore you cross the 2nd half of the room, you need to cross half of the 2nd half of the room. And so forth. At each stage you sub­di­vide the re­main­ing dis­tance in half and cover that dis­tance.

Zeno’s prob­lem was that this se­quence of in­stants is in­finite. If each turn took say 1 sec­ond, it would take in­finity sec­onds to cross the room. From this per­spec­tive any mo­tion—not just room cross­ing—ap­pears im­pos­si­ble. Zeno’s con­clu­sion was that bring­ing in­finity into the dis­cus­sion, in this case di­vid­ing a given dis­tance into in­finitely many pieces, leads to non­sense and should be avoided.

noteFor this and other similar rea­sons, the offi­cial use of in­finity in math­e­mat­i­cal ar­gu­ments was post­poned un­til the in­no­va­tion of Calcu­lus. Essen­tially, due to the si­mul­ta­neous cre­ation of me­chan­ics and all the prac­ti­cal uses to which it could be put, the prob­lems with rea­son­ing with in­finite sums were swept un­der the rug for a while. Calcu­lus uses in­finity all the time and doesn’t apol­o­gize for it. New­ton, Leib­niz, etc. didn’t figure out how to solve Zeno’s para­dox, they just ig­nored it. It wasn’t un­til a cou­ple cen­turies later that math­e­mat­i­ci­ans re­ally con­fronted the prob­lem.

Zeno’s room cross­ing can be writ­ten us­ing num­bers and sym­bols in­stead of sen­tences, where it might look a bit more fa­mil­iar as an “in­finite sum”.


And we can think of its sum ei­ther from the state­ment of the prob­lem (where it is clear that the sum should be 1, a whole room) or by a di­rect calcu­la­tion of the dis­tance re­main­ing to cover and not­ing where its limit is.

Step Dis­tance Covered Dis­tance Re­main­ing
0 0 1
1 \(\frac{1}{2}\) \(\frac{1}{2}\)
2 \(\frac{1}{2}+\frac{1}{4}\) \(\frac{3}{4}\)
3 \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\) \(\frac{7}{8}\)
4 \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}\) \(\frac{15}{16}\)
5 \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}\) \(\frac{31}{32}\)

So, at this point, we have a good rea­son to want to be able to write

$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots = 1$$

Me­ta­phys­i­cal or ex­is­ten­tial doubts notwith­stand­ing, it just looks like we should end up with one if this pro­cess of mak­ing up half the re­main­ing dis­tance were to con­tinue. In this and many other situ­a­tions, its clear that adding to­gether in­finitely many things should work out to give some­thing finite. The way we would read these sym­bols in words is

If you keep adding halves of halves ($\frac{1}{2}$ + \frac{1{4}$ and so on), you get closer and closer to 1. If some­how you could add for­ever, you’d have ex­actly one.

Ex­act and Approximate

At this point we should clear up one thing about ex­act ver­sus ap­prox­i­mate. You might be think­ing that, re­gard­less of the “in­finite” part of the sum, this is a clear case of “good enough for all prac­ti­cal pur­poses”. At some point you get close enough to touch the wall at the end of the room. So yes, we can think of the sum as ap­prox­i­mat­ing one. And these in­finite sums were origi­nally very use­ful be­cause they gave ways of ap­prox­i­mat­ing difficult to calcu­late things with very easy to calcu­late things. But there is some sub­tlety here in the words we use. When we say,

$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots = 1$$

We do not mean

$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots \approx \text{ (is approximately equal to) }1$$

It would be cor­rect to say

$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8} \approx 1$$

or also

$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32} \approx 1$$

but once we have in­finitely many terms, once we are ask­ing what would hap­pen if the pro­cess were al­lowed to con­tinue in­definitely, we are no longer tak­ing about ap­prox­i­mately, we are talk­ing about ex­actly equal.

And this is where peo­ple, not just young stu­dents but the many math­e­mat­i­cal thinkers through his­tory, have had trou­ble. We are used to the things re­lated by equal signs both be­ing sim­ple num­bers. Finite things. Not in­finite pro­cesses. So what does it mean for an in­finite pro­cess to be ex­actly equal to a finite num­ber? So far in this dis­cus­sion we’re not yet sure. But we have a clue. We have ex­actly one case where we have an in­finite pro­cess that has an ob­vi­ous limit. We’d like to equate them.

Next, let’s look at a case where the in­finite pro­cess, in­stead of tel­ling us some­thing we already know, helps us see some­thing new.

Ex­am­ple 2: Archie and the Area of a Circle

You may have been asked to re­mem­ber for­mu­las for the cir­cum­fer­ence and area of a cir­cle in school.

$$ \text{Circumference }(C) = \pi \cdot \text{Diameter }(D) = 2\pi \cdot \text{Radius }(r)$$

$$ \text{Area of a Circle}(A) = \pi \cdot r^2$$

But did you ever won­der why these for­mu­las are true or why they both hap­pen to have \(\pi\) in them? Did you ever won­der how we figured out the area of a round shape like the cir­cle in the first place?

Well, Archimedes did. And the way he figured it out was by cre­at­ing an in­finite sum.

The for­mula for cir­cum­fer­ence is easy. It’s just a defi­ni­tion. An­cient Greeks and many oth­ers no­ticed that \(\frac{C}{D}\) was the same ra­tio for any cir­cle (WHY?). They didn’t call this \(\pi\) or even con­sider it a num­ber, but we do. By defi­ni­tion, \(\frac{C}{D} = \pi\), so of course \(C=\pi D\).

The real mys­tery is with the area for­mula. And this is what Archimedes did. He imag­ined a reg­u­lar poly­gon in­scribed within a cir­cle. Here’s a pic­ture.

![](some pic)

Now, in my pic­ture, there are 8 sides. But there’s noth­ing spe­cial about 8. It could be 6, or 12, or a mil­lion. But…the more sides there are, the more the poly­gon looks like a cir­cle, the bet­ter ap­prox­i­ma­tion its area is for the area of the cir­cle. And, as it turns out, the area of the poly­gon is easy to com­pute. More­over, com­put­ing the area of the py­lon, we can see how it changes with the num­ber of sides and how it stays the same, and we can see what hap­pens to this area if we some­how had in­finitely many sides. Let’s look:

First, we no­tice that a poly­gon can be split into a bunch of tri­an­gles, each of which is the same. How many? The same num­ber as we have sides of the poly­gon. Just like slices of pie. Add up the area of all the slices to get the area of the origi­nal poly­gon.

$$\text{Area of polygon with }n \text{ sides} = \text{sum of } n \text{ triangles}$$

Of course, each tri­an­gle has the same area, and any tri­an­gle has area \(= \frac{1}{2} \cdot \text{base}\cdot \text{height}\). Here, the base is the side of a poly­gon (let’s call it \(s\)) and the height we can call \(h\). Our for­mula for the area of a poly­gon be­comes:

$$\text{Area of polygon with }n \text{ sides} = \underbrace{\frac{1}{2}sh+\frac{1}{2}sh+\ldots+\frac{1}{2}sh}_{n \text{ times}}$$

Now, fac­tor the \(\frac{1}{2}h\) out of the above ex­pres­sion to get some­thing even nicer.

$$\text{Area of polygon with }n \text{ sides} = \frac{1}{2}h(\underbrace{s+s+\ldots+s}_{n \text{ times}})$$

And look, \(n\) sides added to­gether is noth­ing other than the per­ime­ter ($P$) of the poly­gon.

$$\text{Area of polygon} = \frac{1}{2}hP$$

This for­mula is true re­gard­less of how many sides we have. All we have to do now to re­cover the for­mula for the area of the cir­cle is to imag­ine what hap­pens to all of the parts when the num­ber of sides, the ex­tent of the ap­prox­i­ma­tion, the num­ber of terms in the un­der­ly­ing sum, gets larger.

$$ \text{Area of polygon} \rightarrow \text{Area of a Circle}(A)$$

$$h \rightarrow r$$

$$P \rightarrow C$$

So our for­mula be­comes

$$ A = \frac{1}{2}rC = \frac{1}{2}r(\pi D) = \frac{1}{2}r(\pi(2r))=\pi r^2$$

Amaz­ing, no?

There are tons of other situ­a­tions where you can an­swer a ques­tion by putting to­gether in­finitely many pieces, situ­a­tions where we would like to be able to for­mally write down some­thing of the sort:

$$\text{infinite sum} = \text{finite number}$$

but there are also ob­vi­ous and not so ob­vi­ous situ­a­tions where do­ing so is non­sense. We would like to go so far as to ex­tend our rules for alge­bra to be able to op­er­ate on these in­finite sums (things like mul­ti­ply­ing both sides of an equa­tion by the same num­ber or group­ing terms), but the truth is we don’t have enough in­for­ma­tion to know when that does and doesn’t work out. Next we will take a look at this more gen­eral ques­tion.


  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.