Harmonic series and 𝑝-series

for many hundreds of years mathematicians have been fascinated by the infinite sum which we would call a series of 1 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 what it just feels like something that it would be interesting to explore it's 1 over 1 plus 1 over 2 plus 1 over 3 that each of these terms are getting smaller and smaller they're approaching 0 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 and 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 and notice it gets it when it finishes its second full waveform it bends again right at the same time that the wavelength of a ends and then it would be another harmonic would be something that has a third the wavelength of it a the fourth 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 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 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 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 or you have from N equals 1 to infinity of 1 over N to the second power well then it would look like this would be 1 over 1 squared which is 1 and 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 and then you could go on and on for ever 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 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 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 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 harmonic series is a special case where P is equal to 1 this is known as P series so these are known as P series and I try to remember it because it's p4 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 hint it a little bit that maybe some of these converge in 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 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 and we will prove these things in future videos