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!
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
@@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.
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
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.)
@@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 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|
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.
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
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!
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!
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!
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)
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!
@@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).
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!!
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!
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.
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.
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.
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! :)
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.
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. UA-cam lacks great content like yours
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!
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!
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!
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 :)
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 😁
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
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.
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.
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
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
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
I am fascinated about the subject of the video right here, but I am curious of one thing: How did we end up with Digamma of x = H of x *PLUS* 1 - gamma and not *MINUS* 1 at 14:39 ?
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.
Good video but in 10:45 you say a constant function must be the natural choice and anything else should be considered "crazy" but I really can't find a good reason for why it shouldn't be a sinusoidal wave with nodes at the natural numbers which seem to me just as "natural" as it is found in nature.
Pretend he said "simplest" then. Any curve you choose is of course just an approximation. The fact is that you could choose from an infinite class of Taylor polynomials that all look like periodic functions, so in the absence of a reason not to choose the one that's a flat line, you may as well choose the one that's a flat line, since it means your approximation gets to be a polynomial with one term instead of one with infinity terms. As a bonus, the flat line also happens to have a simple derivative (the simplest derivative of all, in fact, since it's 0) which means that extrapolating the method to approximate more points in the interval is extremely simple, instead of forcing you to start over from the beginning every time.
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 :)
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 ❤
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...
10:50 So... WHAT IF... you did use a quadratic between three adjacent points to define your interval. Would you arrive at the same or similar solution? Would they be significantly different or would you be unable to resolve the formula? What cascading effect would this have on the mathematics? And I know I suggested a parabola, but what if we instead used the natural logarithm since they are so related anyway. Wouldnt this be a more "natural curve" since they are shaped so much more similarly? It would be less rooted in intuition and you could make a stronger argument for rigor.
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!
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
keep it up w these kind of videos, rlly good
You definitely deserve it, production quality is what matters in these kinds of explorations
Keep making content man, i will absolutely watch it all... love this kind of videos, already subscribed. Good job! 👍
Are you telling me the points don't follow a logarithm? I'm so disappointed.
Guys, get ready, we are literally witnessing the birth of a new legend in the math-educational UA-cam scene. This is going to be great, i can feel it
i agree
I feel so, too. It reminds me of when I first went crazy for 3b1b
I agree. He has everything 3blue1brown has.
no truer words were spoken.
This aged like milkshake
6:00 that pun was so good! sneaky, unobtrusive, and perfect in context. subscribed.
I didn’t even get it until you said something 😭 thanks
Can someone explain it? I couldn’t get it
@@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.
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?
Thank you!!
I can't really make predictions on how soon, but I definitely have more in the works!
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
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.)
@@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
@@JGHFunRun Well the thing is that this does not hold for any x, but only for x
@@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|
Yeah that part bothered me a bit too
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.
Wau, that's pretty cool
(Another name for the archaic Greek letter "Wau" is "Digamma")
5 nights at wolfram alpha
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
@@therealtdp They probably heard of it from another video, prob not formally studying it but just playing around
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!
Great visualization, great pacing, interesting topic!
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!
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!
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)
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!
@@zanzlanz i'd personally see whether the integral of the geometric series was viable (depending on how expensive non-integer powers are)
@@LinesThatConnect Well, of course you only need to calculate 20. After all, the sum telescopes for positive integer inputs.
@@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).
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!!
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!
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.
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.
Excellent video! Watched through the end, I was basically hypnotized by the quality of the manim animations!!! So cool!!! Keep up the good work
You are a great narrator, I never was excited about sums that much before.
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.
That’s it, I’m binging this entire channel.
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! :)
Beautiful animation, great explanation, fantastic video!
Can't wait to see more of you
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.
This was great. Pretty sure this is the first time I've subscribed to a channel that has only one video!
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. UA-cam lacks great content like yours
I am so glad for 3b1b's Summer of Math Exposition, great videos are popping up everywhere !
Really great video. Good luck with SoME!
That's a great observation about it extending the domain. Seen that formula many times and literally never thought of it that way.
Cannot believe this channel is so underrated. Keep working, and you'll be famous.
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!
Amazing video! Good job, I can’t believe it’s your first one! Keep it up man, I’ll be coming back for more!
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!
I love this kind of "extension" video. Keep them coming
Size does matter, doesn't it?
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!
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 :)
"Just a log"?
Euler-Mascheroni-constant: "Am I nothing to you?"
It's in the name: integral _approximation_
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 😁
Brilliant man, you are not showing math but how to think and investigate thats more important than the math fact itself
Please make more...
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.
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
That thumbnail is perfection
Great video too
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.
Excellent video. Looking forward to future content
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
Nice work. Can’t wait for next episode…
You absolutely need to make more videos like this you are unbelievably amazing ❤
My new favourite channel.
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.
Favourite SoME1 video I've seen so far, really really good video 🥳🥳🥳🥳
When your videos blow up, I can tell everyone, that I was your 14th Subscriber
Great start to your channel. Interesting topic, top class qualify. Subscribed and notification set in anticipation of the next one!
Very well explained. I am amazed about the influence of infinitesimals in modern math.
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
Yeah, but for integers, the fractions cancel until you're left with the original definition.
This is amazing, and I'm really excited to see you're upcoming videos
Holy shit! This video is amazing and surprisingly enough this is your first video... 3blue1brown quality level
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
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
First SoME1 video I could actually understand!
I am fascinated about the subject of the video right here, but I am curious of one thing: How did we end up with Digamma of x = H of x *PLUS* 1 - gamma and not *MINUS* 1 at 14:39 ?
Обожаю после просмотра заходить в Desmos и смотреть, как работают эти формулы :D
You deserve more subscribers, this video is extremely well made!
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.
Absolutely amazing video!
this is now my favorite math video!
You're a legend. Kudos to you!
Good video but in 10:45 you say a constant function must be the natural choice and anything else should be considered "crazy" but I really can't find a good reason for why it shouldn't be a sinusoidal wave with nodes at the natural numbers which seem to me just as "natural" as it is found in nature.
Pretend he said "simplest" then. Any curve you choose is of course just an approximation. The fact is that you could choose from an infinite class of Taylor polynomials that all look like periodic functions, so in the absence of a reason not to choose the one that's a flat line, you may as well choose the one that's a flat line, since it means your approximation gets to be a polynomial with one term instead of one with infinity terms. As a bonus, the flat line also happens to have a simple derivative (the simplest derivative of all, in fact, since it's 0) which means that extrapolating the method to approximate more points in the interval is extremely simple, instead of forcing you to start over from the beginning every time.
Finally someone who make video about this topic. And that method is awesome 👌
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.
Amazing video! Extremely clear explanation and a very well chosen topic. Simple yet extremely slick arguments. Subbed!
Great video, best math content I've seen in a while
Wicked animations. Loved the video!
How do you only have 1k, this deserves way more attention.
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 :)
This was a joy to watch.
Next step: generalize it to complex numbers
Then find all the zeros
The same formula works
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 ❤
the digamma function works in the complex plane its just harder to compute
What the sigma notation
Great video! Looking forward to seeing more from this channel.
I'll be waiting for the next one just to hear that outro again
Can I just say, the thumbnail is godlike
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
In a couple of years I will be happy to say that i was your 210th subscriber
You did a very good job with this video, nice work man.
Great video! Really informative! I’m excited to see what you do in the future!
Imagine going to infinity + 0.5 only to come all the way back to determine the precise value of H(0.5). Amazing!
Wonderfully explained, math is so beautiful. Looking forward to your new content, you'll surely make it big!
That's such a cool trick to understand intuitively, you made it very simple. I hope you plan to make more videos like this!
I'm SOOOOOOOOOOOOOOOOOOOOOOO in love bro, i love this, this came handy to motivate me learn those series i have a course on
Wonderful!! More videos, please! :)
This is great content, I hope your channel grows!
That is something that I had been wondering for a long time
Just found you today and I honestly loved the video, I hope to see more in the future.
That limit function really put a smile on my face.
THOU SHOWED ME THE TRUE BEAUTY OF MATHEMATICS!!!!!😊
This video is beautiful!
This is an excellent video, well presented and well explained. I'm looking forward to your future videos.
Fantastic. I subscribed, can't wait for more!
Very nicely done. Looking forward to more!
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...
This thing was so dope. You sir are really cool
This video was great! On par with (or maybe even better) than a 3blue1brown vid!
Neat👍😎
Amen to that
Amazing presentation, looking forward to more stuff!
What a great video! Enjoyed every second
10:50 So... WHAT IF... you did use a quadratic between three adjacent points to define your interval. Would you arrive at the same or similar solution? Would they be significantly different or would you be unable to resolve the formula? What cascading effect would this have on the mathematics?
And I know I suggested a parabola, but what if we instead used the natural logarithm since they are so related anyway. Wouldnt this be a more "natural curve" since they are shaped so much more similarly? It would be less rooted in intuition and you could make a stronger argument for rigor.
This war very well done! Please continue making these nice math videos! 👌
Def one of the better math channels in youtube. keep it up! it would be cool to do some computer science videos :)