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## AP®︎/College Calculus BC

### Unit 10: Lesson 5

Harmonic series and p-series

# Harmonic series and 𝑝-series

AP.CALC:
LIM‑7 (EU)
,
LIM‑7.A (LO)
,
LIM‑7.A.7 (EK)
𝑝-series is a family of series where the terms are of the form 1/(nᵖ) for some value of 𝑝. The Harmonic series is the special case where 𝑝=1. These series are very interesting and useful.

## Want to join the conversation?

• Is there a difference between p-series and the Riemann Zeta Function?
• Yes, their technical definitions make them significantly different from each other, although it may not seem so at a glance.

Any given p-series is defined as
``f(p) = Σ[n=1→∞](1/n^p)``

where `p` is a real-valued variable.
The Riemann zeta function, on the other hand, is defined as
``ζ(s) = Σ[n=1→∞](1/n^s)``

where `s` is a complex-valued variable.

This difference between the p-series function being real-valued and the zeta function being complex-valued is critical, because it means that we can only use real analytic (i.e., calculus with real numbers) techniques with p-series functions but we can use complex analytic (i.e., calculus with complex numbers) techniques with the Riemann zeta function, which waters all sorts of sprouts of research in the field of analytic number theory to this day (whereas large interest in p-series kind of died out when Leonhard Euler solved the Basel problem in the 18th century), mostly in the form of the Riemann hypothesis, widely considered to be the greatest unsolved problem in pure mathematics.
• I am struggling to make sense of the fact that the series at and are converging/decreasing series. I am unable to make sense of the fact that the terms are decreasing from 1, and have to compute the LCMs of two terms, make them like fractions, and then observe which of the two is greater.
Could somebody please suggest what topics of basic math I should brush up on, so that I fill gaps in my lack of understanding, and understand p-series and related concepts better?
(1 vote)
• There's a much more efficient way than using LCM's here, because the numerators are all the same. Fractions with the same positive numerator and increasing positive denominators form a decreasing sequence of fractions, so we can see immediately that the terms are decreasing. This is a basic concept of fractions in arithmetic. If a pizza is sliced into a greater number of equal pieces, then each piece becomes smaller.

You may need to brush up on both arithmetic and algebra, and focus on developing good mathematical intuition about the effects of operations on numbers. To understand series, you will also need to have enough intuition to distinguish conceptually the terms themselves from the sequence of partial sums of the terms. It's very easy to fall into the trap of assuming that a series must converge just because the terms are getting smaller.
• should'nt the harmonic series converge to 1?
• The harmonic series is greater than 1 after the first two terms, and only increases after that.
• Is 1/(3n) also a harmonic series?
(1 vote)
• The harmonic series is the exact series 1+1/2+1/3+1/4...
There are no others. 'The harmonic series' is the name of one particular series, not a class of series.

However, 1/(3n) is one-third of the harmonic series (at any partial sum), so it diverges as well.
• Does the p-series (which Sal introduces at ) have a general term formula like an arithmetic progression does?
(1 vote)
• The nth term of a p-series is 1/(n^p), but there's no nice general formula for the partial sum of the first n terms of a p-series.
• Is a diverging series one whose terms become increasingly larger (that is, become larger and larger as the series goes on)?
(1 vote)
• Not necessarily! A divergent series is a series whose sequence of partial sums does not converge to a limit. It is possible for the terms to become smaller but the series still to diverge! In the situation of the p-series, the terms have to shrink fast enough in order for the series (sequence of partial sums) to converge instead of growing without bound.
(1 vote)
• What does "p-" stand for; is it just used in reference to this series's general form using customarily the letter "p"?
(1 vote)
• p just represents power series

edit incorrect, I think it is rho the greek letter not p.
(1 vote)
• should the harmonic series always be in the form of fractions?
(1 vote)
• Is p explicitly an integer here? Or does this property of convergence when p>1 and divergence when p≤1 work for any real p?
(1 vote)
• A p-series will converge for all real p > 1, not just integers greater than 1.
(1 vote)
• I have problem 1/(n^1/4 +3) and I was wondering if I could still use the p-series to show if it converges or diverges?
(1 vote)

## Video transcript

- [Instructor] For many hundreds of years, mathematicians have been fascinated by the infinite sum, which we would call a series, of one plus 1/2 plus 1/3 plus 1/4, and you just keep adding on and on and on forever. And 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 hit 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, and what's true about the harmonics are that they will be 1/2 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 and notice, it gets, when it finishes its second full wave form, it ends again right at the same time that the wavelength of A ends. And then it would be another harmonic where it'd be something that has 1/3 the wavelength of an A and a 1/4 of a wavelength of A, and if you look at a lot of musical instruments or what sounds good to our ears, they're playing not just a 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, harmonic series. And in a future video, we will prove that, and I don't want to ruin the punchline, but this actually diverges, and 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 one to infinity of one 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. Doesn't hurt to rewrite it and get more familiar with it. This right over here is the harmonic series. One over one, which is just one plus one over two plus one over three, 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 one to infinity of one over n to the second power. Well, then it would look like this. It'd be one over one squared, which is one, and we can just write that first term as one, plus one over two squared, which would be 1/4, plus one over three squared, which is 1/9, and then you could go on and on forever. 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 one to infinity of one 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 one plus one over two to the p plus one over three to the p plus one over four to the p, and it doesn't just have to be an integer value. It could be, some, p could be 1/2, in which case, you would have one plus one over the square root of two plus one of the square root of three. This entire class of series and of course, 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 'cause it's p for the power that you are raising this denominator to. You could also view it as you're raising 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 one, 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's 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 and we will prove these things in future videos.