Harmonic series and 𝑝-series | AP®︎ Calculus BC | Khan Academy

For many hundreds of years, mathematicians have been fascinated by the infinite sum which we would call a series of one plus one-half plus one-third plus one-fourth, and you just keep adding on and on and on forever.

This is interesting on many layers. One, it just feels like something that would be interesting to explore. It's one over one plus one over two plus one over three; that each of these terms are getting smaller and smaller, they're approaching zero. But when you add them all together, these infinite number of terms, do you get a finite number or does it diverge?

Do not get a finite number. This also shows up in music, and this actually might have been one of the early motivations for studying this series. Where if you have a fundamental note, a fundamental frequency in music, and the point of this video isn't to teach you too much about music, but if you have a fundamental note that might be a pure A or something like that, I'm just showing you one of its wavelengths.

Obviously, you would keep going like that, and this is a hand-drawn version so it's not perfect. The harmonics are the frequencies, the overtones that at least to our ear reinforce that A. What's true about the harmonics are that they will be one half of the wavelength of A, in which case it might look something like this.

So this would be a harmonic of A; it has half of the wavelength of A. Notice it gets it when it finishes its second full waveform. It ends again right at the same time that the wavelength of A ends. Then it would be another harmonic, which would be something that has a third the wavelength of A, fourth of a wavelength of A.

If you look at a lot of musical instruments or what sounds good to our ears, they're playing not just the fundamental tone but a lot of the harmonics. But anyway, that was a long-winded way of justifying why this is called the harmonic series.

Harmonic on harmonic series, and in a future video, we will prove that, and I don't want to ruin the punch line, but this actually diverges. I will come up with general rules for when things that look like this might converge or diverge, but the harmonic series in particular diverges.

So if we were to write it, so in sigma form, we would write it like this: we’re going from n equals 1 to infinity of 1 over n. Now, another interesting thing is, well what if we were to throw in some exponents here?

So we already said, and I'll just rewrite it; it doesn't hurt to rewrite it and get more familiar with it. This right over here is the harmonic series: 1 over 1, which is just 1 plus 1 over 2 plus 1 over 3, so on and so forth. But what if we were to raise each of these denominators to say the second power?

So you might have something that looks like this, where you have from n equals 1 to infinity of 1 over n to the second power. Well then it would look like this: it would be 1 over 1 squared, which is 1. We could just write that first term as 1 plus 1 over 2 squared, which would be 1/4, plus 1 over 3 squared, which is 1/9.

Then you could go on and on forever, and then you could generalize it. You could say, “Hey, all right, what if we wanted to have a general class of series that we were to describe like this?”

Going from n equals 1 to infinity of 1 over n to the p', where p could be any exponent. So for example, well, the way this would play out is this would be 1 plus 1 over 2 to the p plus 1 over 3 to the p plus 1 over 4 to the p. It doesn't just have to be an integer value; it could be some p could be one-half, in which case you would have 1 plus 1 over the square root of 2 plus 1 over the square root of 3.

This entire class of series, and of course, the harmonic series is a special case where p is equal to one. This is known as p series. So these are known as p series, and I try to remember it because it’s p for the power that you are raising this denominator to.

You could also view it as erasing the whole expression to it because one to any exponent is still going to be one. But I hinted a little bit that maybe some of these converge and some of these diverge, and we're going to prove it in future videos.

But the general principle is if p is greater than 1, then we are going to converge. And that makes sense intuitively because that means that the terms are getting smaller and smaller fast enough.

Because the larger the exponent for that denominator, that means that the denominator is going to get bigger faster, which means that the fraction is going to get smaller faster. And if p is less than or equal to one, and of course when p is equal to one, we're dealing with the famous harmonic series, that's a situation in which we diverge. We will prove these things in future videos.