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# Proof: harmonic series diverges

AP.CALC:
LIM‑7 (EU)
,
LIM‑7.A (LO)
,
LIM‑7.A.8 (EK)

## Video transcript

this is a painting of Nicole oh Ram I looked up a phonetic spelling before making this video I'm assuming I'm still mispronouncing it but and I print out but I apologize to all the French speakers out there ahead of time but he was a famous French philosopher mathematician who lived in medieval France he lived in the 1300s and he's famous for his proof that the harmonic series actually diverges and just as a little bit of review this is the harmonic series 1 plus 1/2 plus 1/3 plus 1/4 plus 1/5 and it's always been in my brain the first time that I saw the harmonics here it wasn't obvious to me whether it converged or diverged it looks like well gee you know all these terms were positive but they're going towards zero so I could imagine that that this thing could converge but he proved otherwise he proved one of the most famous and most elegant proofs in mathematics that it does indeed diverge and the way that he did this is he replaced every term in the harmonic series with a term that is less than or equal to that term and then if he proves and if he if he if he could if he and then by proving that his new series diverges and it's less than or equal to this series or each of the terms are less than or equal to each of the corresponding terms of here then he says therefore by the comparison test this must diverge so how did he construct that well one way to think about it is he replaced each of the terms in the harmonic series with the largest power of 1/2 that is less than or equal to that term so what's the largest power of 1/2 that is less than or equal to 1 well one is a power of 1/2 so that is 1/2 to the 0 power is 1 so 1 is the largest power of 1/2 that is less than or equal to 1 so I'll just write the 1 there and now what's the largest power of 1/2 that is less than or equal to 1/2 well that's just going to be 1/2 that's just 1/2 to the first power now what's the largest power of what's the largest power of 1/2 that is less than or equal to 1/3 well one half is larger than one-third it's not less than one-third we want it to be less than one-third so the next power of 1/2 is or I should say the power of 1/2 the largest power of 1/2 that is less than or equal to one-third is 1/4 so replace 1/3 with 1/4 and of course replace 1/4 with 1/4 and then we get to 1 and then we get to 1/5 what's the largest power of 1/2 that is less than or equal to 1/5 once again 1/4 is greater than 1/5 the largest power of 1/2 that is less than or equal to 1/5 is 1/8 so you replace that with 1/8 of course would replace for the same reason that with 1/8 you would replace that one with 1/8 and of course 1/8 the largest the largest power of 1 of 1/2 that is less than or equal to 1/8 is 1/8 and then what would he replace 1/9 with well he would replace 1/9 with one 16th by the exact same argument and you'd keep going all the way until you get to 1/16 so you would essentially have 8 1/16 in a row well what's interesting here well 4 let's first verify that we can we can use the comparison test here so in this first series each of the terms are non-negative in the second series each of the terms are non-negative and we also see that each of the corresponding terms in a harmonic series is larger than each of the core be greater than or equal to the corresponding terms in this years we constructed it this way these are equal this one is equal this is greater than this 1/3 is greater than 1/4 1/4 is equal to 1/4 1/5 is greater than 1/8 1/6 is greater than 1/8 1/7 is greater than 1 8 1 8 is equal to 1/8 and so one way to think about it is this is the each of the corresponding terms and this new constructed series are smaller and I'm going to just call it s in this infinite sum and of course we keep going on and on and on maybe I should do that in magenta so we see that each of the corresponding terms here are smaller than the corresponding terms up here and they're all positive so if we can prove if we can prove that s that this sum right over here diverges then by the comparison test the larger series the harmonic series here the one where the corresponding terms are larger that must also diverge and how do we do that well let's just actually just take these sums this is going to be so let me write it so s is going to be equal to 1 plus 1/2 1/4 plus 1/4 what's that what's 2/4 or 1/2 I think you see what's going on here this is exciting what's 1/8 plus 1/8 plus 1/8 plus 1/8 well that's 4 8 or 1/2 what's 1/16 plus 1/16 and we're going to go all the way until we get to 1 although we're going to have 8 of these so that's going to be 8 16 or 1/2 1/2 and then you're going to have 16 1 30 seconds or 1/2 and so we're essentially just going to be adding 1/2 and we start with the 1 we just keep adding what plus 1/2 plus 1/2 plus 1/2 plus 1/2 well this is clearly going to be equal to or this is going to this is unbounded this is you could say that this is equal to infinity this is equal to infinity or another way to think about it is s clearly diverges s clearly diverges and since s is I guess we could say each of its corresponding terms or each of its terms are smaller than the corresponding terms in the harmonic series we can then say that the harmonic series diverges our monic series diverges diverges there's no way that this thing over here can converge if this thing is each of its corresponding terms are smaller and you could even think of this kind of the sum as being smaller but this sum goes to infinity so this one must also go to infinity anyway hopefully you found that as interesting as as I did
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