The Harmonic Series and your gut feeling
There’s no feeling like being completely misled by your own intuition, and the harmonic series is one of those little things that gets the job done.
If you take the following expression:
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ….
You may wonder what happens as you continue doing this, ad infimum.
The thing you are adding to the right keeps getting smaller, until it becomes negligibly small. You may be wondering if it still contributes something to the total value. What happens to this total value?
This expression is called the “harmonic series”. The N-th term of the harmonic series is the sum of the first N reciprocals (a “reciprocal of x” is a fancy way of saying 1/x).
Thus, the fifth term (N=5) is
1 + 1/2 + 1/3 + 1/4 + 1/5 = 2.28333…
So, what does the hundredth, thousandth, millionth, billionth, etc term look like, and do they converge to one value, or not?
Let’s calculate these:
N = 5 → 2.28333…
N = 10 → 2.92897…
N = 100 → 5.18738…
N = 1000 → 7.48547…
N = 1000000 → 16.69531…
N = 1000000000 → 21.30048…
As you can see, this grows very, very slowly. After a billion terms you only get to something like 21.3004.
After that its growth slows down more and more.
It actually takes 15092688622113788323693563264538101449859497 terms to finally cross the value of “100”.
So, where does it go, in the end? Does it “stall” at a single value, or does it keep growing?
Let’s look at some other series just to get a feeling if it is even possible to stall to one value with an infinite addition of numbers.
For instance, let’s look at the squares of the reciprocals.
1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + …
I’ll call it the “squared series” here — it doesn’t really have a name, I think.
The things you add as you go on, are even smaller than with the harmonic series.
And, in fact, this series does converge, to the value π²/6 (= 1.644934).
This is known as the “Basel problem” and established Leonhard Euler as the greatest mathematician in the highly competitive region since he was the one to solve it — with very simple tools.
If you want to know more about solving this, check out this thread [https://math.stackexchange.com/questions/8337/different-methods-to-compute-sum-limits-k-1-infty-frac1k2-basel-pro].
Now, you may be surprised by the appearance of “π (= 3.14…)”, and it is a common mistake to just glance over this without taking the opportunity to let the mystery of “π” completely sink in.
There are a lot of “reasons” why π² appears here, and there’s not a single answer.
This is not a cop-out, it’s more like trying to say why water is blue. Water isn’t blue in the first place: the sky is blue, and this in itself has as much to do with your own eyes as with complicated physics of the electromagnetic force and beyond.
Ultimately all of these are connected to the same truths about the universe, but there are several ways of connecting these truths into an explanation.
That is one facet of truth, after all, that it can be re-established in various ways from its consequences.
My favourite “version” to explain why π appears here (and especially π squared), is explained by the amazing 3blue1brown here.
Definitely watch it if you have the time!
Basically, an “infinitely long line” problem is converted into the problem of an “infinitely large circle” (which looks like a straight line, after all).
While the length of the circle and its diameter become infinitely large, their ratio stays the same: π. Now, the way 3blue1brown assembles it, the “squared series” can be represented by something that uses the square of this ratio, and then it comes out to have π squared when you look at the infinite case.
Now, the fact that the “squares” series
1 + 1/4 + 1/9 + 1/16 + 1/25 + …
converges to “something” at all, is fairly easy to understand without resorting to some more complex mechanics.
You can first look at a different series, which I will call the “semi-factorial” series
1 + 1/2 + 1/6 + 1/12 + 1/24 + …
Where the denominators follow the progression 1, 2, 6, 12, 20, … which is
1, 1x2, 2x3, 3x4, 4x5…
and notice two things:
First, it’s bigger than the squares series, since the denominators of the square series are always bigger so the sum of reciprocals is smaller. (Fact 1)
Secondly, you can rewrite this new series as
1 + (1–1/2) + (1/2–1/3) + (1/3–1/4) + (1/4–1/5) + …
Now, just by cancelling the terms, you can see that this ultimately becomes 2. (Fact 2)
So, by combining Fact 1 and 2, you establish that the “squared series” becomes smaller than 2. Since you also know that it never becomes negative, you can tell that this means it converges to “something” positive smaller than 2.
Now, let’s try the same technique on our harmonic series. What we’re going to be doing here is what was actually done by (probably) Nicole Oresme at least 600 years ago, although it was only later re-discovered that he did it.
Let’s first write it out to a few terms.
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + ….
We can rewrite this as:
(1) + (1/2) + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + …
Now, each term in between parentheses is bigger than 1/2. So, the whole thing, the whole infinite series is bigger than something that is created by just adding 1/2 ad infimum.
Well, adding 1/2 ad infimum to something is surely going to become infinitely large. So, the harmonic series will also become infinitely large.
Now, I think there is a deep problem with the above reasoning, and I’ve never felt satisfied with it. I think it is confusing to children.
The thing is, if you look at the amount of terms you need to group into parentheses, for the last set we already need four terms:
(1/5 + 1/6 + 1/7 + 1/8)
For the next group, we will need eight terms, sixteen, thirty-two, and so on. Ultimately, as we stretch this into infinity, we will need an infinite amount of terms in the last group to make our point that this group is larger than 1/2. This may feel problematic to many people to get their head around — it does, to me.
Luckily there are other tools that can be used to prove that the harmonic series diverges. Usually these tools involve mapping the problem to more traditional “continuous” mathematics.
Since the harmonic series clearly grows at a pace of 1/N (each result for N is 1/N bigger than the last one), it is highly reminiscent of the natural logarithmic function, which grows at a pace of 1/x as well (a pace that keeps slowing down as x gets larger and larger).
The natural logarithmic function is the function that represents which power of e you need to get a number x.
If “e” confuses you here, it’s not that important for now: the reasons works for powers of “10” as well. And, that “10” logarithmic function evolves slowly through 0, 1, 2, 3, 4 … as x hits 1, 10, 100, 1000, 10000, …
You definitely know it by now through all of the Covid-19 graphs that have been published, that need to resort to logarithmic scales to deconstruct the “feeling” of a certain growth — a straight or bent line in logarithmic scales can be more meaningful than a curve in normal scales.
Now, while the pace of growth of the logarithmic function slows considerably, you still need an infinite power of 10 to represent something “infinite”, so it never settles down.
The harmonic series is like a “jagged” brother of the logarithmic function, only taking “e” instead of “10” as its exponent because it is a little lower in all its terms. Also it would be much more crazy to have the number “10” appear here than the number “pi” or “e” because the only reason why we use the number 10 is for biological reasons — we have ten fingers. So, it comes out that using “e” as the base of the logarithmic function makes them come together.
This jagged companion follows its smooth sister, very slowly, upwards, towards infinity. It takes over 10⁴³⁴ terms for it to hit 1000. Deep terms are actually difficult to calculate because of the limited precision used by computer calculations.
It’s not very intuitive that it does forever increase, is it?
Now, there are three bizarre facts about the harmonic series that may stretch this newfound intuition even further.
The Euler-Mascheroni constant
First, looking at the jagged brother and its smooth sister, they track each other, but not precisely. In fact, there’s a difference between them, that settles to a specific number, as you go on into infinity.
This number is the Euler-Mascheroni constant, and it is 0.5772156649…. (The decimals keeps going on).
It is one of mathematics’ most unresolved mysteries whether this number is irrational or even transcendental (transcendental meaning, whether it can be the solution of some equation that involves powers of x and nothing more, irrational meaning, whether it can be expressed as a complex fraction).
Many mathematicians think we will never solve this problem with our current tools!
The Euler-Mascheroni is a rather unintuitive number that appears in many outcomes.
It seems to say something about the “impact” of the granularity of natural numbers as they grain against against the continuity of real numbers (sorry if I offend anyone with this handwaving but it is how it feels like to me).
It is unclear if some of the more “peculiar” numbers in the physical universe (for instance the fine structure constant) have some relation to it — this is a typical statement that might aggravate physicists but to me I’d rather hope they are fundamentally linked.
The Alternating Harmonic Series
A different bizarre fact about the harmonic series is the alternating harmonic series.
1–1/2 + 1/3–1/4 + 1/5–1/6 + 1/7 — …
Now, rather strangely, this series does converge (to ln 2, which is the power y you need for e^y = 2), by being “pushed” by vertical forces into this value.
Now, what may be non-intuitive by this however, is, that if you rearrange the terms, you can actually change the outcome.
If for instance you rewrite
1–1/2 + 1/3–1/4 + 1/5–1/6 + 1/7 –…
As
1–1/2–1/4 + 1/3–1/6–1/8 + 1/5–1/10–1/12 + …
And calculate out this pattern, even visualising the pattern on a graph as above, the sum will come out as half of what it was before, all of a sudden. Note we didn’t leave out any terms, we just rearrange them (1/7 will come up a bit later in the series now)
In fact, it’s possible to rearrange the alternate harmonic series in such a way that you can represent any number with its infinite sum. Just the arrangement of terms will end up carrying a meaning to the final result. This is precisely why I was always a little wary of Nicole Oresme ’s proof of the divergence of the harmonic series.
Don’t trust your intuition when it comes to infinity!
The One with the Missing Numbers
Something insanely unexpected happens if you “exclude” some of the numbers appearing in the harmonic sequence.
Let’s look at what happens if we exclude every number that has a “3” in its denominator.
So, instead of doing
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + …
We do:
1 + 1/2 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + 1/11 + 1/12 + 1/14 + …
So, we excluded the terms 1/3 and 1/13, because they have a “3” in their denominator.
Then, what happens if we do this ad infimum?
Well, in that case, the series does converge!
In fact, if we leave out numbers according to any pattern at all (whether we leave out numbers that have “4” in them, or numbers that have the string “5876846” in them, any pattern), we peel off enough numbers so that the harmonic series no longer diverges to infinity. In fact, they quickly converge to very low numbers that way.
If you want to know more about this, look at the Kempner Series article on MathWorld [https://mathworld.wolfram.com/KempnerSeries.html]
I would just love for you to sit and think about this for a short while: if we leave out all the numbers of the harmonic series, that have the number “989078748629” in their denominator — whatever number you can think of! — you leave out enough terms to make the series no longer grow to infinity.
As you can see its growth towards infinity was always quite fragile.
Never trust your own intuition in these fragile circumstances!
