The Strange Music of the Harmonic Series

part 2 in a finite series on infinity

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?

It isn’t.

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:

28 thoughts on “The Strange Music of the Harmonic Series”

1. But is it just 9? Surely same applies to 8, right? Or 7, 6, etc. And if it converges, what does it converge to?

1. Roy says:

You are right, and it will be impossible to calculate the number it converges to. I wonder if the number of terms is finite or infinite, even though the probability for a term to be “useable” in the series tends to 0%?

1. BlueHairedMeerkat says:

The number of terms must be finite; say your string to avoid is just 9; then a substring of your allowed sequence is 1,11,111,1111…, which is infinite, so the whole sequence must be infinite.

1. BlueHairedMeerkat says:

Aaaarg. Infinite, rather. That’s supposed to say infinite.

2. Reblogged this on mhorley and commented:
The Harmonic Series diverges, but remove the 9s and it converges. But what number does it converge to?

3. The resulting series is called the Kempner Series and it’s been shown that (at least to 20 decimals) the sum is 22.92067661926415034816

4. If we imagine a function f(n)=C, where n is integer string that is excluded from the series and C is the resultant convergent sum, what would a graph of n vs. C look like? Somebody with enormous amounts of free time should draw one.

1. Shouldn’t take too long (he says!) One for the summer holidays…

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2. Andrey says:

whoa… The graph won’t give you much, but… whoa…

3. Joseph says:

If I’m interpreting the paper correctly, if n is an m-digit number, then f(n) will be pretty close to (10^m)*(ln 10). So, if n has 1 digit, then f(n) is roughly 23. If n has 7 digits, then f(n) is roughly 23 million. Etc…

5. vlorbik says:

cool post. i was kind of surprised at this phenomenon when a grad student told me about it a year or two ago (using 6’s rather than 9’s)… but he gave me a good clear explanation and now it feels like it should’ve been obvious. kind of like the monty hall problem in this way.

6. pete mack says:

Just saw your post on metric time. There are a few old style measures that make sense.
1. American units of volume from TBSP to Bushel are in powers of two. This is awesome for cooking, much more useful than powers of 10.
2. Degree as unit of angle. 360 is a really useful thing, since with grads you don’t have 30-60-90 triangles. You have 33 1/3, 66 2/3 100 triangles. From this you get:
Nautical mile. 1 arc minute at the equator
The same logic holds for a 24 hour day: it is easier to divide into equal parts. So you can have 8 hour shifts or four hour watches or 6 hour quarters etc. With a 10 hour day, you are stuck with halves and fifths. That’s not nearly so useful. So for units that need to be easily divisible, get a few factors of two and all the early primes. Otherwise, stick with powers of 10.

7. I’m unconvinced. I believe you of course but my mind won’t get around it.

Here’s what my mind is telling me (It gets fooled a lot):

While the series without 9’s is slowing to a crawl, even infinitely slowing, it never slows to 0. Wouldn’t it have to slow to 0 for their to be a boundary? There is always a number without 9’s that you can add no matter how small of a chance their is of finding it.

Choose as many digits as you like, isn’t one of them always just 1’s going on infinitely? This proof looks at it and says there is a great likely hood that any number with a vast amount of digits has a nine in it and I look at it and say it’s also a certainty that you will eventually find a number without any 9’s in it. What gives?

1. Yeah, infinity is where our intuitions go to war. Not all can survive!

In this case, the intuition that perishes is your assumption that if I keep adding more terms, the sum will get bigger and bigger and bigger, growing without bound. This isn’t true.

Just look at 1/2 + 1/4 + 1/8 +…

By your thinking, this one should also grow to infinity. But it never exceeds 1.

Weird terrain.

1. Agree completely your series converges but we’ve given it a definition so that it converges. I’m looking (and my brain thingy is looking too) at the non convergent harmonic series
1+1/2+1/3+1/4..

and comparing it to that same series simply without any number containing one million 9’s in a row. The rule we’ve created sure stunts it’s growth, particularly near infinity, but my brain isn’t convinced that it converges at any point because there is always something to add to it no matter how small. My brain’s infinity just isn’t big enough.

I’m almost convincing myself now that the harmonic series must converge somewhere if almost any rule based subset of it converges.

My real problem could be that if you take every ninth number out of the harmonic series it still diverges but if you take all the 9’s out it converges.

Trickery and witchcraft I tell ya.

but I think I’m almost there….

8. Brilliant!
I especially loved these lines :

“I am thinking of gigantic numbers like 238 trillion.

Exactly. Small numbers. “

9. Err, so if we throw out the nines, what number does it converge too?
*Runs off to see if Excel is big enough to calculate anything meaningful*