Extending the Harmonic Numbers to the Reals

[@LinesThatConnect]

Thanks for watching everyone! I obviously hoped to see a positive response, but I didn't think I'd actually get thousands of views on my first video! It makes me really happy to see so many people enjoying this topic. The Harmonic Numbers are the tip of an iceberg into some really cool math, and I hope to eventually share that whole journey on this channel. I've been working on another (completely unrelated) video, but as I'm doing this between work and other personal projects, I can't give any estimates as to when it will be ready. I just wanted you to know that there definitely is more on the horizon!

[@CMDRunematti]

oh yeah you gonna blow up. havent watched these kinds of videos in a while and youtube just puts you on my home page...pretty sure youre gonna have a huge spike of views (or just had and the algo still loves you) been a pleasure to watch too

[@Rugjoint]

keep it up w these kind of videos, rlly good

[@arunchebrolu1113]

You definitely deserve it, production quality is what matters in these kinds of explorations

[@victorvalencia6466]

Keep making content man, i will absolutely watch it all... love this kind of videos, already subscribed. Good job! 👍

[@leofigoboh1611]

Are you telling me the points don't follow a logarithm? I'm so disappointed.

[@sebastiangudino9377]

Guys, get ready, we are literally witnessing the birth of a new legend in the math-educational KZbin scene. This is going to be great, i can feel it

[@warpedmine9682]

i agree

[@spongebobbatteries]

I feel so, too. It reminds me of when I first went crazy for 3b1b

[@roiburshtein852]

I agree. He has everything 3blue1brown has.

[@Deadslinger-Doshita]

no truer words were spoken.

[@khallell]

This aged like milkshake

[@eammonful]

This was excellent. I've watched a lot of the SOME1 videos and this is easily one of the best. Do you expect to release any more anytime soon?

[@LinesThatConnect]

Thank you!! I can't really make predictions on how soon, but I definitely have more in the works!

[@minyiiiii]

6:00 that pun was so good! sneaky, unobtrusive, and perfect in context. subscribed.

[@cooliostarstache6992]

I didn’t even get it until you said something 😭 thanks

[@vancedforU]

Can someone explain it? I couldn’t get it

[@minyiiiii]

@@vancedforU he refers to the "real magic" if the function because it extends the harmonics from the whole numbers to the real numbers. just well placed.

[@FareSkwareGamesFSG]

When I first learned about sigma notation and the Riemann Zeta function, I spent the next 5 nights playing on Wolfram Alpha, and it was fun. It was very similar to what you've done here! Thanks for the connection with the digamma function, too, I never knew what that was.

[@diceLibrarian]

Wau, that's pretty cool (Another name for the archaic Greek letter "Wau" is "Digamma")

[@RichConnerGMN]

5 nights at wolfram alpha

[@therealtdp]

how did you go from learning the sigma notation to learning the zeta function so fast? i learned sigma notation like 4 years ago, and i AM sure, that i wont be studying zeta function for atleast the next 3 years

[@bishan_8617]

@@therealtdp They probably heard of it from another video, prob not formally studying it but just playing around

[@鸞麤]

Nice video. One very small point I would make would be at 10:55 on your justification of using a straight line for the interval, as opposed to some squiggly up and down one. I believe it would be better to justify it under the intuition that the harmonic functions extended to the reals should be a strictly increasing function, rather than the rather loose "it is the most natural. It's probably being overly pedantic, but mathematics is all about rigour

[@ery5757]

Even if you add the argument of strict increase, it is not necessary for the function to be a straight line, there still is some intuition to it. The argument probably should be that if f(x) does not change much also every derivative of f goes to 0. And even this is hard to proof as you can not do the limit of the difference quotient because you do not have infinetely close points for which the function is defined. Anyways, as we are looking for a function that behaves kind of natural in some way, we can just take that as a definition of "natural": that every derivative disappears on the interval between arbitrarily close points. (Now as I am writing this I wonder, if you could proof that under the condition, that every derivative should be continous in any point - but I am not enough of a mathematician to try that myself.)

[@JGHFunRun]

@@ery5757 well for the really big numbers in order for it to be monotonically it needs to be close to a straight line, otherwise it would go above the second value/below the first. It may not be exactly but as you get bigger and bigger it gets closer and closer. This is expressed by the statement that for any x lim[N->inf] H(N) - H(N + x) = 0

[@ery5757]

@@JGHFunRun Well the thing is that this does not hold for any x, but only for x

[@JGHFunRun]

@@ery5757 yea um x was (heavily) implied to be finite when N explicitly goes to infinity. You’re being extremely pedantic and I’m not even sure why - for any finite x, x inf] N since N >> inf. Heck in the video he explicitly said something equivalent to “for very large N in relation to x” N>>x. Do you think the statement of “very large N in relation to x” stopped applying at some point? Because if so tell me where it stopped applying. I’ll wait Oh, and if you really want to be pedantic it’s only if |x|

[@schizoframia4874]

Yeah that part bothered me a bit too

[@jackrookstool1306]

Dude, this video is one of the best produced math videos I’ve seen in a while! You have the elegant animations of 3Blue1Brown while also touching subjects I’m more interested in, so bravo, honestly! Keep up the good work, I’m excited to see what you have in store in the future!

[@DrZye]

Great visualization, great pacing, interesting topic!

[@danidotexe_]

Got excited that I'd found a new cool maths channel. Got disappointed when I realized it's only your first video. Got excited again when I realized it's only your first video. Please keep it up!

[@zanzlanz]

Amazing first video - I can't wait to see what's next! Not taking into account the reasons why you'd prefer one notation over another, I think it's a bit curious though how computing the first numbers gets more and more difficult as you get further into this video - I mean at the start it is just evaluating fractions, and by the end you have to calculate an infinite sum!

[@LinesThatConnect]

Thank you!! The infinite sum is great because you can take derivatives and integrals, and in general a bigger domain means more potential to find interesting patterns. But if you ask me for H(20), there's no chance I'm adding infinite terms when I only need to do 20! (Plus, the sum converges pretty slowly, especially for larger numbers)

[@zanzlanz]

Oh! Can you can do calculus on sums? I always thought that was kind of a dead end! It's still pretty nuts how do calculate the Nth harmonic, you have to add the first N reciprocals. Coming from a computer science background, I feel the best way to implement this would not be one of the mentioned formulas; a lookup table is too sparse, the classic definition gets too slow too quickly, and the summation takes too long to converge. I'm kinda baffled by how many ways there are do calculate this one thing!

[@Errenium]

@@zanzlanz i'd personally see whether the integral of the geometric series was viable (depending on how expensive non-integer powers are)

[@angelmendez-rivera351]

@@LinesThatConnect Well, of course you only need to calculate 20. After all, the sum telescopes for positive integer inputs.

[@tomkerruish2982]

@@zanzlanz You can do calculus on sums; you just have to make sure there aren't any shenanigans going on. It's like with mixed partial derivatives; usually they're equal, unless the function is going nuts at some point. (For more eye-glazing details, look up the Monotone Convergence and Dominated Convergence Theorems.) As for calculating harmonic numbers, there are asymptotic expansions which work pretty well. For example, H(n) is about ln n + gamma + 1/2n - 1/12n² + 1/120n⁴, where gamma is the Euler-Mascheroni Constant. You can continue extending this approximation, but unfortunately for any finite n the terms start getting larger after some point (this is a general problem with asymptotic series). However, the computational advantages are enormous. Finding the billionth harmonic number by summation would take literally billions of calculations and only give you about eight places of accuracy past the decimal point; the above formula (with a previously calculated value of gamma) yield over fifty places after the decimal point (the first omitted term is of order 1/n⁶, or one part in 10⁵⁴).

[@peepoclown1]

Wow, I rarely find a new channel to which I’d like to subscribe, but I’ve «never» been so quick to hit the bell icon too. 0: Extraordinary success lays just over the horizon for this channel. Keep it up!!

[@hemanthkotagiri8865]

I didn't even notice 15 minutes have gone by. That's how good you are at explaining things. Awesome, man! Keep going! The world needs more lucid explainers like you!

[@ЧингизНабиев-э2г]

Damn, I am a high schooler who is interested in maths. Gamma-function, Digamma function, harmonic numbers and extension of series from integers to real(and complex) numbers are definitely one of my favorite topics. Honestly, this is the best video about harmonic numbers I’ve seen so far.

[@WolfrostWasTaken]

Excellent video! Watched through the end, I was basically hypnotized by the quality of the manim animations!!! So cool!!! Keep up the good work

[@isavenewspapers8890]

13:05 I think it sheds some light if we apply this formula to obtain a value we already knew previously. For instance, we can plug in x = 3 and then expand the series: H(3) = (1 / 1 - 1 / 4) + (1 / 2 - 1 / 5) + (1 / 3 - 1 / 6) + (1 / 4 - 1 / 7) + ... We see that the -1 / 4 cancels with the 1 / 4, the -1 / 5 will cancel with a 1 / 5, and so on. So we're just left with 1 / 1 + 1 / 2 + 1 / 3, exactly as expected.

[@1SLMusic]

That’s it, I’m binging this entire channel.

[@avz1865]

Wow! I used essentially this idea to find a continuous extension for the factorials! I know the gamma function already exists, but doing it this way gives you a different formula that happens to give you exactly the same thing as the gamma function. I never considered doing the same thing with other functions, so cool!

[@FlynnFromTaiga]

You are a great narrator, I never was excited about sums that much before.

[@mahxylim7983]

12:28 Math is crazy in a way that you can have an good intuition about every separate fact but when combined, knocks you back to the start and realize you don't really comprehend the whole story.

[@aaronrainbolt3154]

I was legitimately getting worried when you started using approximations and was like "uh, this can't work can it?", then you brought a limit into the picture and it was like a light switch went on in my brain. That was awesome.

[@somniad]

This is excellent! And the first video on this channel? This sort of thing has a sizeable audience, I'm sure, and it's distinct from what I'm seen elsewhere. Keep it up!

[@mandresyfalimanana3538]

I'm SOOOOOOOOOOOOOOOOOOOOOOO in love bro, i love this, this came handy to motivate me learn those series i have a course on

[@michaelriberdy475]

I love this kind of "extension" video. Keep them coming

[@YoutubeModeratorsSuckMyBalls]

Size does matter, doesn't it?

[@a52productions]

It was hard for me to silence the voice in my head screaming "It's just a natural log! Use an integral approximation!", but this was definitely worth it! Great video :)

[@landsgevaer]

"Just a log"? Euler-Mascheroni-constant: "Am I nothing to you?"

[@danielyuan9862]

It's in the name: integral _approximation_

[@richardfredlund8846]

That's a great observation about it extending the domain. Seen that formula many times and literally never thought of it that way.

[@telnobynoyator_6183]

I am so glad for 3b1b's Summer of Math Exposition, great videos are popping up everywhere !

[@benjaschunk3461]

Next step: generalize it to complex numbers

[@moshadj]

Then find all the zeros

[@nickfaire]

The same formula works

[@モ.イブラヒム]

To actually see it try plotting it in geogebra it's just like desmos but I personally love it more just type psi(x+1) -psi(1) and it will graph it for you ❤

[@PowerUpStudio_]

the digamma function works in the complex plane its just harder to compute

[@Best-Amphetamine]

What the sigma notation

[@davutsauze8319]

Hi, I have watched nearly all of 3blue1brown's videos, and yet I still think yours was one of the best I have ever encountered. I am *begging* you to upload another one, you could easily match 3b1b's videos in quality, and you in fact already did, if not better. KZbin lacks great content like yours

[@OrigamiCreeper]

Damn this video is so impressive! Especially for the first video, it feels like the product quality matches 3b1b. I look forward to your future videos!

[@martindaza6222]

I'm usually a casual viewer when it comes to math videos but man... videos like this makes you appreciate how beautiful math is. Really cool video, hoping to see more! :)

[@rysea9855]

Beautiful animation, great explanation, fantastic video! Can't wait to see more of you

[@debaratighosh3866]

Please make more videos like this. I want to know the mystery behind why graphs look the way they look for particular equations. This is amazing

[@esquilax5563]

This was great. Pretty sure this is the first time I've subscribed to a channel that has only one video!

[@05degrees]

Nice! Though we *can* get H(0) = 0 from the original sum-of-reciprocals definition too: for that case the sum has no terms, and an empty sum equals zero (likewise an empty product equals one). Don’t be afraid that the sum is “from 1 to 0” in that case-that’s equivalent to being from 1 inclusive to 1 exclusive, which then makes more sense to have zero terms in it.

[@isavenewspapers8890]

So by the definition you're using, does the lower bound need to be exactly 1 less than the upper bound for the expression to be defined this way? I've seen a definition whereby the upper bound being any number less than the lower bound causes the entire sum to evaluate to 0. In that case, then H(-1) would also be 0, contradicting the result in the video.

[@05degrees]

@@isavenewspapers8890 Yeah I guess exactly like that, one 1 less than the other! (I already forgot the context of that video but this makes sense.) In general there indeed exist different contexts where any upper bound less than a lower bound will make an empty set to sum or integrate over, but also contexts where the sum or integral will just change their sign, and that inclusive-exclusive business can make some things hairier that they are but in the end we can sort contexts of both kinds apart (the second thing usually happens in algebraic settings and when signed, say, differential forms can be presumed, which is why ∫[a; b] = −∫[b; a], and the first thing happens probably when we can’t do this, I don’t remember exactly...)

[@isavenewspapers8890]

@@05degrees I imagine you're referring to the extension whereby \sum_{k=a}^{b} f(k) = -\sum_{k=b + 1}^{a - 1} f(k) for a > b, which can be derived by recursively applying the formula \sum_{k=a}^{b - 1} f(k) = -f(b) + \sum_{k=a}^{b} f(k). I mean, I did this myself, so I'm hoping I didn't screw up. Anyway, I guess it's similar to integrals, where you can flip the sign and swap the bounds. However, there's a slight adjustment to avoid off-by-one errors. I guess that's what you're describing as "inclusive-exclusive business"; honestly, I'd have expected to hear that in a computer science context rather than a math one. I've just never thought of sum notation in that way. I'm actually fascinated by your statement of the existence of integrals where you have nothing to sum up because the upper bound is less than the lower bound. It runs so counter to all the experience I've had with integrals, which is presumably in the "algebraic context" you speak of. If you remember where these kinds of integrals pop up, please let me know.

[@05degrees]

@@isavenewspapers8890 I wrote it badly, sorry. I don’t know about integrals per se that do this thing. What I’ve seen though is defining a segment of integers [m..n] so that when m > n it’s an empty set. And then when we define a sum of an integer-indexed sequence, we can use a definition of a segment like this, making sums from m to n zero if m > n. I’m not sure generalizing this kind of behavior to integrals is useful but _maybe_ somebody indeed have done that.

[@isavenewspapers8890]

@@05degrees Oh, I see. Thank you for the explanation.

[@andriypredmyrskyy7791]

I discovered this while "discovering" the stamp collector's problem for myself. There's a fun approximation that's useful if you're trying to find how long it takes to get items from a random draw.

[@ERROR-ei5yv]

As someone who's interested in somewhat niche generalizations like this, this video was really interesting! It was very well explained and visualized and easy to understand

[@lltheguy8387]

I love your teaching style! I hope you enjoyed making this video as much as I enjoyed watching it, cause if you keep up this level of quality, you WILL find success here 😁

[@woowybaby3064]

Amazing video! Good job, I can’t believe it’s your first one! Keep it up man, I’ll be coming back for more!

[@TheJara123]

Brilliant man, you are not showing math but how to think and investigate thats more important than the math fact itself Please make more...

[@emma5068]

On paper, I knew how to derive this infinite sum. But this video did a fantastic job of making it much more intuitive. Well done.

[@Gooey]

That thumbnail is perfection Great video too

[@adaminsall9713]

Really great video. Good luck with SoME!

[@モ.イブラヒム]

You absolutely need to make more videos like this you are unbelievably amazing ❤

[@okkoheinio5139]

Great video! One thing I would have liked to see would be showing that the infinite sum you arrived at indeed gives the values of the harmonic series when x is an integer

[@danielyuan9862]

Yeah, but for integers, the fractions cancel until you're left with the original definition.

[@timurpryadilin8830]

Cannot believe this channel is so underrated. Keep working, and you'll be famous.

[@AKAIMAX1]

Great video! Honestly this was presented so well I decided to go to wiki, and start deriving some of the stuff that was presented there as well as following the steps that were taken in this video. Really well done :)

[@nicolasyan1613]

This is a really good video! The pacing, narration, and animation are all very smooth and pleasant to watch. There's one piece of feedback I have, though this may be more a matter of my personal taste, so take it at your discretion. It's natural for someone who's familiar with maths to understand which of the arguments in the video are rigorous proofs and which are just natural-looking assumptions. However, I'm not sure that would be clear to everyone. I think it would be good to really underline the point that there's an infinite number of possible functions which connect the dots, but that _if_ we enforce the recursive relation and _if_ we assume the curve flattens out, _then_ we get the final function.

[@Adityarm.08]

nit : 1:20 : A cleaner (imho) formulation for binary exponent sum is if we start from k=0. Then it's just 2^(n+1)-1. This is extremely insightful into the exponential nature of how the next term is just one added to the sum of everything that came before.

[@RaymondBarbour]

Great start to your channel. Interesting topic, top class qualify. Subscribed and notification set in anticipation of the next one!

[@joedasilva134]

Very well explained. I am amazed about the influence of infinitesimals in modern math.

[@benheideveld4617]

Nice work. Can’t wait for next episode…

[@overyx]

Обожаю после просмотра заходить в Desmos и смотреть, как работают эти формулы :D

[@Archer-bc6cv]

This is amazing, and I'm really excited to see you're upcoming videos

[@SeggieSum]

You deserve more subscribers, this video is extremely well made!

[@ianprado1488]

Excellent video. Looking forward to future content

[@pendragon7600]

dude you should have won the contest. I've seen all the winning videos and there's some steep competition but this should have been top 5.

[@raunakmukherjee8029]

Amazing video! Extremely clear explanation and a very well chosen topic. Simple yet extremely slick arguments. Subbed!

[@upsilonalpha3982]

This video was great! On par with (or maybe even better) than a 3blue1brown vid!

[@daveederer5553]

Neat👍😎

[@alexakalennon]

Amen to that

[@Vearru]

That limit function really put a smile on my face.

[@tomspray5961]

Favourite SoME1 video I've seen so far, really really good video 🥳🥳🥳🥳

[@eugenet453]

I'll be waiting for the next one just to hear that outro again

[@schmunzelndKatze]

"But before we go on, I'd like to introduce--" *eyes glaze over and prepares to skip the next part* "--Another form of the recursive formula" *Snaps back to reality*

[@Singularidade]

Holy shit! This video is amazing and surprisingly enough this is your first video... 3blue1brown quality level

[@nickyblosser1345]

Just found you today and I honestly loved the video, I hope to see more in the future.

[@RedDaLord]

That's such a cool trick to understand intuitively, you made it very simple. I hope you plan to make more videos like this!

[@alaindeturing62]

Great video, best math content I've seen in a while

[@hiiistrex2838]

Can I just say, the thumbnail is godlike

[@RubyPiec]

First SoME1 video I could actually understand!

[@Caspitein]

Wonderfully explained, math is so beautiful. Looking forward to your new content, you'll surely make it big!

[@3of7tricom34]

Great video! Really informative! I’m excited to see what you do in the future!

[@klaus9356]

This is great content, I hope your channel grows!

[@bogdanmarandiuc2895]

Thank you, i've been wondering for like half a year why it's limit goes to infinity and finally i found the answer ty

[@devsquaredTV]

Def one of the better math channels in youtube. keep it up! it would be cool to do some computer science videos :)

[@Cr42yguy]

Imagine going to infinity + 0.5 only to come all the way back to determine the precise value of H(0.5). Amazing!

[@Corncycle]

Great video! Looking forward to seeing more from this channel.

[@hierkonnteihrewerbungstehe5636]

When your videos blow up, I can tell everyone, that I was your 14th Subscriber

[@brighamhellewell6479]

this is now my favorite math video!

[@gabedarrett1301]

This is insanely good! Subscribed! Somewhat unrelated, but I have a video idea: proving that some functions have non-elementary antiderivatives and thus lack an analytical solution. Just throwing it out there... :)

[@שוןאלמליח]

Absolute brilliant video, I love those exploration videos that take you through a journey of discovery. And this video does it perfectly. Can’t wait for the next video

[@exoplanet11]

Thanks for a cool, well made video. It is great to see high quality math pedagogy on channels like this one and 3brown1blue. When I saw the strange behavior for x

[@Kapomafioso]

This is exciting! One solution to "extend the harmonics" I came up with was to notice how 1/k would be a result of integrating x^(k-1) from 0 to 1. But how do we use this? Well, we write 1+1/2+1/3+...+1/n = integral (1+x+x^2+...+x^(n-1)) from 0 to 1. But what do we do with this? Then I realized, that there's a nice algebraic formula with a telescoping effect (try to expand the left-hand side yourself to see how almost everything cancels out!) (1+x+x^2+...+x^(n-1))(1-x) = (1-x^n) so H(x) = integral (1-t^x)/(1-t) for t from 0 to 1. I'm not sure if it gives the same values between the integers as your formula tho...to me, both extensions seem equally logical. I like my solution because it's easy to set it up as a numerical integral. I tried the value of 1/2, which is both easy to integrate and easy to find the sum for and the result is 2-ln(4) in both cases, so...yay? :D

[@fatih3806]

I put them both on Desmos (infinite sums don’t work so I used sum until 10000 for their sum which is big enough to be almost same for all small values of x) and they put out the same values everywhere. Except yours doesn’t give any values for x smaller than -1

[@ColeCoug]

Wicked animations. Loved the video!

[@mategart1766]

Finally someone who make video about this topic. And that method is awesome 👌

[@wafikiri_]

So interesting, clear and smart, I've subscribed. Thanks! I'm not very involved in mathematics, although I acquired a senior engineering level half a century ago. But now I am involved in a theory of cognition and have had to re-learn lattices, ideals and filters. Unfortunately, I cannot find much material on the subject in KZbin.

[@voodooguru]

This is an excellent video, well presented and well explained. I'm looking forward to your future videos.

[@bazwardo7191]

How do you only have 1k, this deserves way more attention.

[@adnanahmad2549]

THOU SHOWED ME THE TRUE BEAUTY OF MATHEMATICS!!!!!😊

[@benterrell9139]

My new favourite channel.

[@MasterZeroYTTM21]

You did a very good job with this video, nice work man.

[@zazinjozaza6193]

This is very cool. A few years ago I asked myself this very same question and used a completely different method to find the solution which ended up leading me nowhere. I ended up googling the result. Funny to see you make this video, I feel a strange kinship...

[@NoNTr1v1aL]

Absolutely amazing video!

[@complex314i]

[1+2+3+...+n]^2 = [.5n(n+1)]^2 = .25n^2(n+1)^2 = 1+8+27+...+n^3 Cubing then summing is equal to summing then squaring. Such a strang yet fascinating result.

[@TagetesAlkesta]

I plugged the formula into Desmos on my phone and it crashed lol

[@MathyJaphy]

Very nicely done. Looking forward to more!

[@perrydimes6915]

that synthwave beat made me so hype to see the episode on the gamme function

[@xyz.ijk.]

That was very well done. I'm looking forward to going back and watching more of your other videos.

[@aioia3885]

That is something that I had been wondering for a long time

[@nemesida4166]

Это настолько интересно, что я не хочу идти в школу после просмотра этого шедевра. Спасибо)

[@claytonbacon7730]

i love math and i’m so high this was amazing

[@edyt4125]

Lmfao

[@gc-foi-espoir-amour]

Wonderful!! More videos, please! :)