A Fake Proof That The Harmonic Series Converges

[00:00:00]Hello everybody.

[00:00:02]So let's do another fake proof of a fake proposition.

[00:00:05]So in the last video we proved that every person on Earth has the same name.

[00:00:10]Okay, well there had to have been a mistake in that fake proof. In this one we're going to fake proof that uh the harmonic series converges.

[00:00:20]So if you took calculus then you might remember that if you add up 1 + 1/2 + 1/3 + 1/4 and so on that is divergent.

[00:00:29]That sum eventually gets above 100. If you go far enough out, if you go far enough out it gets above a million, if you go far enough out it gets above a billion, a trillion, any number you want, eventually this sum it gets above it. It is diverging off to infinity.

[00:00:41]Very, very slowly, but it is. Uh in my uh math history book I include a lot of the history of it. Uh it's very interesting and some of the uh early proofs of the fact. Anyways, so yeah, it it it diverges. So the fact that I'm saying that this sum is less than infinity, meaning convergent, that is false. It's false. That's a that's a that's a fake result. Uh but yet I'm going to prove it to you.

[00:01:06]>> [laughter] >> Well, okay, we'll see. See So this is your challenge for this video. See if you can identify where in the proof I have an error. Okay. So here we go. We proceed by induction. Let S sub n be this statement. Okay, so I'm using the notation from uh a few videos back when I wrote down um the statement of of of induction, right?

[00:01:29]How does induction work? You have to have a sequence of of mathematical statements. You have to prove that the first one is true. You have to prove that the k-th one implies the k plus first one.

[00:01:39]And then you know they're all true. So S sub n is just the n-th statement. So here's the n-th statement. The sum from 1 + 1/2 + 1/3 all the way up to + 1/n.

[00:01:48]So we stop at 1/n. That sum is less than infinity. Okay? So that's the n-th statement.

[00:01:54]So what's our base case?

[00:01:57]Our base case we have to show that S of one is true, right?

[00:02:01]So the base case is if n equals one then we have we're stopping right at one, right? We're stopping before even get going.

[00:02:13]Then one is less than infinity. That is a true statement.

[00:02:19]Um so S of one is true.

[00:02:24]Okay, so the base case is satisfied because if you stop at one, certainly you're you're convergent. I mean there's nothing to converge. It's it's a sum of one thing.

[00:02:33]Um okay, what about the inductive hypothesis? Well, this is the statement that if you choose a k in the natural numbers and assume that we assume that S of k is true, what does that mean? It means the sum from one plus a half plus a third is stopping at one over k is less than infinity, okay? So we sum up to one over k and then we stop.

[00:02:54]Um so that's convergent. That's a convergent sum.

[00:02:58]Um okay, so that's the inductive hypothesis. Now for the induction step.

[00:03:03]Um so whenever you prove something by induction, when you get to the induction step, the most important question to ask yourself is how am I going to use the inductive hypothesis to prove this?

[00:03:15]So by the inductive hypothesis what do we want to do? We want to sum up to k plus one over k plus one as the last term, right?

[00:03:25]We're to sum one plus a half plus a third up to one over k plus one. One step before that was one over k. So that's how we're going to use it.

[00:03:34]So um so this by the inductive hypothesis um summing up one plus a half plus a third all the way up to one over k is some number.

[00:03:52]Some finite number.

[00:03:56]Call it F.

[00:03:57]Okay? So, we have some finite number uh that it equals the sum of the first K.

[00:04:03]Um Actually, in the book I called it F. So, let's switch it to F. Just in case you're following along.

[00:04:10]Um so, we have F there.

[00:04:12]Um then what happens when you add one more term?

[00:04:17]Well, 1 + 1/2 + 1/3 up to 1 over uh >> [snorts] >> K + 1 is equal to um the sum One step before this is 1 over K.

[00:04:42]And the sum up to K equals F. Some finite number.

[00:04:48]So, that equals F + 1 over K + 1.

[00:04:54]And here I mean, nothing mysterious is happening. We have F + 1 over K + 1. A number plus a number is some other number, right? This is some finite number. Right?

[00:05:05]Um equals this which is finite.

[00:05:10]Right? It's the sum of two finite things.

[00:05:15]So, it is less than infinity.

[00:05:20]Okay.

[00:05:23]Some shorthand there, but it's less than infinity, right? So, this sum is equal to this, which is less than infinity.

[00:05:29]Okay?

[00:05:31]So, we've proven the induction step.

[00:05:35]And now, therefore, the conclusion right? It holds in the infinite case, too.

[00:05:44]So, by induction Um this means that the infinite sum is also less than infinity.

[00:06:07]Uh completing the proof.

[00:06:10]So, that's it.

[00:06:19]So, I'll put a end of proof box, but I'm a little shaky about it, so I'll make it kind of a wiggly one.

[00:06:26]It's a QED box is kind of trembling a little bit.

[00:06:30]>> [laughter] >> I'm sure about it's itself. Um So, what do you think?

[00:06:36]It is not true that this sum is finite. It is infinite.

[00:06:42]But if it's infinite, there has to be a mistake somewhere in this argument.

[00:06:46]So, um why don't you take a look at it and see if you can find the error?

[00:06:55]And I think I'll do what I did last time and say if you find the error, if you think you have it, why don't you leave it in the comment section on YouTube?

[00:07:04]And uh I think probably someone will be able to identify it and write it up well.

[00:07:10]And I'll comment on on that on that reply. Um if no one does, then I will go ahead and make another video explaining it more fully, but I suspect uh my very intelligent and good-looking YouTube audience will be able to do it. Always compliment the people who might give you a rating and review.

[00:07:31]>> [laughter] >> Okay, so uh give that a shot, see if you can figure out where the error lies in the logic in of this proof, why this is not a a proof by induction, there's some error somewhere. And if you can figure it out, leave a comment. And and yeah. Uh check that out. Uh next time we will begin talking about the back matter of uh of the of the chapter. We are wrapping up chapter four of the proof book.

[00:07:58]Uh the next section is bonus examples and then there's you know exercises and stuff. So uh there's pro tips and stuff.

[00:08:03]So I'll talk about that uh next time. Okay? See you then.

[00:08:11]>> [music] [music]