part 2 in a finite series on infinity
(see also part 1)
A few weeks ago, the webcomic Saturday Morning Breakfast Cereal posted a cartoon about the harmonic series.
(Obviously it’s a mistake to post an actual cartoonist’s work alongside my own second-grade-quality scrawl, but hey, maybe I’ll benefit from a math humor cheerleader effect.)
Now, what is the harmonic series? It’s this:
The sum never stops. It goes on forever and ever. Lovely, yes, but does it—in any meaningful sense—“equal” anything?
That is to say: Does it level out at some value?
Or does it just keep rising, forever and ever, eventually exceeding a million, then a billion, then a trillion, and so on, surpassing any ceiling or limit we might imagine?
Perhaps your first thought is this: “You silly man; you’ve already answered your own question. You said the sum goes on forever. So it must be infinite.”
Not so fast.
Take a look at this series, for example:
We start at 1. The next term brings us halfway to 2. The next term brings us halfway again to 2. The next term brings us halfway AGAIN to 2.
And so on.
This sum is “infinite” in the sense that it goes on forever, with no final term. But it’s finite in the sense that no matter how many numbers you add, you’ll never exceed 2. Sure, you’ll get close—achingly, painfully, infinitesimally close—but you’ll never surpass it. We say that the sum “converges” to 2.
Now, back to our original sum, the harmonic series. What should it equal?
Well… let’s look at some running totals as we go along.
Where is it going to settle down?
To be clear, this total grows incredibly slowly. In fact, the word “slowly” fails to evoke its agonizingly incremental pace. Instead, imagine that you’re waiting in line at the DMV.
And there’s only one employee.
And that employee is one of those talking trees from Lord of the Rings.
And he’s stoned.
And the line includes all 7 billion people on earth.
That line you’re in? It moves like ball lightning compared with the growth of this series.
And yet… this series never settles down. This sum eventually exceeds a million, then a billion, on its way to the stars. In mathematical language: the series diverges.
This is all weird enough. But now we get to the truly strange part of the whole ordeal, the truth that prompted the inimitable Zach Weinersmith of SMBC to build a punchline around it:
If you throw out the numbers with 9’s in them, the series is small enough to converge.
“Nonsense, you gullible old toad!” you are perhaps shouting to your screen. “Why should throwing out the numbers with 9’s make such a difference? We’ve still got all the other numbers!”
Again, I say: not so fast. You’re making a classic mistake. When you think of “numbers,” you’re only picturing little numbers.
“No I’m not!” you may say. “I’m thinking of big numbers. Huge numbers. Like 9 million, or 47 billion, or 228 trillion.”
Exactly my point: small numbers.
You see, the longer a number gets, the harder it is to avoid a 9. Every time you add a digit, you add a new opportunity for a 9. That might not feel like a big danger—after all, those 9’s will pop up only 10% of the time. But look what happens:
It’s as if, every time you type a digit, there’s a 10% chance that a giant numeral “9” falls from the sky and crushes you. For short numbers, with just 2 or 3 digits, you’re not very worried. The chances are in your favor.
But now you begin to type a 100-digit number. How do you like your chances? Sure, that giant “9” probably won’t fall on this digit… nor on this digit… but how long do you think your luck will last? Eventually you’ll get unlucky, and that “9” will come plummeting from the sky.
In the long run, most numbers have 9’s in them. Virtually all of them, in fact. So if you throw out these numbers from the harmonic series, it’s no surprise that it now converges. You’ve thrown out almost the entire series!
Want to know the weirdest part? The same logic applies to a longer sequence of digits. Say, 999.
At first, most numbers won’t have this sequence. But picture a billion-digit number. Just writing this number out—in size 8 font, double-siding the printing to save trees—takes a stack of paper as tall as a house.
Surely somewhere in there the number is bound to have the digits “999” in that order, right?
The same is true of every equally big number. And MOST numbers are this big! After all, there are only finitely many smaller numbers, and infinitely many bigger numbers. So, again, we’re throwing out almost the entire series.
Thus, the harmonic series also converges if you throw out all the numbers that include the digits “999.”
Perhaps you can see where this is going. (If so, you may be experiencing vertigo and/or nausea; this is normal.) The logic above works for any string of digits you can possibly think of. Even, say, a sequence of a million 9’s in a row.
So, in conclusion, the harmonic series diverges. It eventually outgrows any ceiling you’d put on it.
But simply throw out the numbers that happen to include a string of a million 9’s in a row… and suddenly the series converges. It plateaus. There’s some value that it will never surpass.
Somehow, by excluding only those numbers with a million 9’s in a row, you’ve changed the nature of the series.
In the words of Weinersmith: