Topic 10.5: Harmonic Series and p-Series

The endless crowd

It’s the last song of the night, the guitar’s resting on its stand, and the singer asks everybody out there to hold up their phones. One by one the whole field lights up, tens of thousands of tiny lights, and every single one of them sends a little bit of its light back to this stage.

Now let’s imagine something impossible. Imagine this crowd never ends, ring after ring of fans going out forever, and ask yourself whether the light on this stage would ever stop piling up. Then let’s take the very same endless crowd and line everybody up in one single file, stretching straight out from the stage.

One of these two crowds pours an infinite amount of light onto this stage. The other one has infinitely many fans in it too, and it can’t even manage twice the light of the very first fan.

Today you’ll prove which crowd is which. Then you’ll do the same for much uglier endless sums, the kind the AP exam loves to throw at you. Before we find out, make your call. Which crowd do you think floods the stage with infinite light?

Why it’s called harmonic

To figure out why, we need one sum, and it’s been hiding in that guitar the whole time. A plucked guitar string doesn’t only swing as one big piece. It also wiggles in smaller pieces at the same time, and you can make those ring on their own. If you touch the string lightly, right at the middle, it rings in two halves, each one half the length of the string, and the note jumps up an octave.

Touch it a third of the way along and it rings in three pieces, each a third of the string, and a quarter of the way along gives you four quarters. Musicians call these the harmonics. Their lengths give us one, a half, a third, a quarter, and so on, and when we add them up we get the harmonic series, the sum behind last lesson’s leaning books.

Let’s copy one loop from each harmonic and lay the copies end to end across the stage, the whole string first, then a half, then a third, then a quarter. The ribbon gets past two string lengths after four pieces, past three after eleven, and past four after thirty-one, so it’s still growing, but it’s slowing down a lot. Let’s put a finish line ten string lengths away. How many pieces do you think the ribbon needs to get past it?

It takes twelve thousand three hundred sixty-seven pieces to get past ten. If a computer had added a trillion pieces every second since the Big Bang, the ribbon would still only be about sixty-nine string lengths long.

Last lesson the integral test proved that this ribbon never stops, but there’s an older proof, from around thirteen fifty, that you can actually watch with nothing but these pieces. Look at pieces three and four, a third and a quarter. Each one is at least a quarter. If we trim both down to a quarter, they snap together into exactly half a string, and the real pair is even longer.

Now you do the next group. Trim pieces five through eight down to the length of the shortest one, then snap them together. Each of the next eight pieces is at least a sixteenth, so those eight pieces give us another half. Every group has twice as many pieces as the one before, so another half is always waiting. Infinitely many halves add up to more than any number you can name, so the ribbon passes every finish line you could draw.

On the board, it’s the sum from n equals one to infinity of one over n, one plus a half plus a third plus a quarter, and so on. The terms go to zero, but as you saw last lesson, that alone can’t make a series converge, and this series is the reason why.

Let’s write that old proof down. Group the terms, a third plus a quarter, then a fifth through an eighth, then the next eight, and so on. Now shrink every term in a group down to that group’s smallest term. That can only make the sum smaller, so the real series is at least as big as this.

Every group is now exactly a half. That means the first four terms add up to at least one plus two halves, and the first eight to at least one plus three halves. In general, the first two to the k terms add up to at least one plus k halves, and that grows without bound. That means the harmonic series diverges, even though its terms go to zero.

Notice the move we just made, because it’s the second half of this lesson. We compared our series with a smaller series that we already knew runs off to infinity, so ours had to run off too.

Let’s use the groups once more. One thousand twenty-four is two to the tenth, so at least how big is the sum of the first one thousand twenty-four terms?

Before we leave the guitar, let’s make one small change and flip every other piece around, so it’s forward one whole string, back a half, forward a third, back a quarter. The hops keep shrinking and switching direction, so the bead zigzags in tighter and tighter and closes in on one spot, about sixty-nine percent of the way along the string.

That’s the alternating harmonic series, one minus a half plus a third minus a quarter, and unlike the harmonic series it converges. Its sum is the natural log of two, and next lesson you’ll prove why it settles down.

The line and the rings

Our crowd has been adding up these very same sums, so let’s go back and call the light from the first ring of fans one. The second ring is twice as far around, so it holds twice as many fans. Each of those fans is also twice as far away, and light from twice as far away is four times dimmer.

Twice the fans, at a quarter of the light each, gives you half the light of the first ring. The third ring has three times the fans, each nine times dimmer, which gives a third, and in general ring n gives one over n.

That means the rings add up to the harmonic series, and we just proved that it grows forever. Turn them all on and the light on the stage keeps on climbing, with no ceiling at all. In the single line the fans stand a meter apart, so fan number n is n meters out and sends one over n squared. The sum becomes one plus a quarter plus a ninth plus a sixteenth, and so on.

Watch the meter. It climbs fast at first, then slows way down, and however many fans we add, it never gets past about one point six four five, less than twice the first fan. The only difference between the two crowds is the power on n, one for the rings and two for the line, and that power decides between infinite light and a ceiling.

Let’s find out exactly where the switch happens. This slider thins the crowd out between full rings and a single line, so slide it around and find the thinnest crowd that still floods the stage with infinite light.

Every sum like these has the same shape, one over n to some power p, and we call it a p-series. A p-series converges when p is greater than one and diverges when p is one or less, and that’s the whole test. For any positive p, one over x to the p is positive, continuous and decreasing, so last lesson’s integral test applies.

The antiderivative is x to the one minus p, over one minus p, so the integral from one to infinity of one over x to the p, d x, is finite exactly when p is bigger than one, and the series follows the integral. You saw this picture last lesson for p equals two. The rectangles after the first one sit under the curve, so the whole sum is less than one plus the integral, which is two. That’s exactly why the endless line can’t double the first fan.

The cliff at one is razor sharp. With p equal to one point zero one, the integral, one over p minus one, comes out to a hundred. The sum sits between the integral and one more than the integral, so it’s between a hundred and a hundred and one, while at p equals one it’s infinite.

Some p-series come in disguise, so always rewrite the term as a single power of n first. The square root of n over n squared is n to the one half over n to the second, which is one over n to the three halves. Three halves is more than one, so that one converges.

Watch out for an n on top, though. The sum of n over n squared looks like p equals two, but it simplifies to one over n, which is the harmonic series, so it diverges. Your turn. Which one of these four series do you think converges?