Power sum MASTER CLASS: How to sum quadrillions of powers ... by hand! (Euler-Maclaurin formula)

[@Mathologer]

The longest Mathologer video ever! 50 minutes, will this work? Let's see before I get really serious about that Kurosawa length Galois theory video :) Today's video is another self-contained story of mathematical discovery covering millennia of math, starting from pretty much nothing and finishing with a mathematical mega weapon that usually only real specialists dare to touch. I worked really hard on this one. Fingers crossed that after all this work the video now works for you :) Anyway, lots of things to look forward to: a ton of power sum formulas, animations of a couple of my favourite “proofs without words”, the mysterious Bernoulli numbers (the numbers to "rule them all" as far as power sums go), the (hopefully) most accessible introduction to the Euler-Maclaurin summation formula ever, and much more. Also, the channel recently hit 500K subscribers. Thank very you to all of you for your support :) After all this time doing everything by myself I am getting a little bit tired of the non-math side of things (editing, subtitles, etc.) and I am thinking of enlisting some professional help. Anyway, to this end I've now switched on the least annoying type of KZbin ads and I am thinking of finally putting up a Patreon page. What sorts of things would you like to see there? As usual thank you very much to my friend Marty Ross for nitpicking this one to death (especially for not letting off until I finally inserted that "morph" shortcut in chapter 7 :) Finally, check out the article "Gauss’s Day of Reckoning" by Brian Hayes which tells the story behind the famous story of Gauss adding 1+2+3+...+100 as a kid: tinyurl.com/y49buyak Finally, finally, typos at kzbin.info/www/bejne/nKiUnIWwbZh5oJI and kzbin.info/www/bejne/nKiUnIWwbZh5oJI (x on top should be 1, of course. I thought I'd fixed this one, but apparently not :(

[@srijanchakraborty9137]

That was a wonderful video. I understood most of the things except 41:01 I'm in class 12

[@brucefrizzell4221]

I do not understand all of it , but these are very interesting things to study next . I find an exploration of my ignorance to be a rewarding journey . Thank you for working so hard to bring them to my attention.

[@ruinenlust_]

I understood all of it except 41:01... please set up a patreon!

[@kristianthulin]

This is perfect! I have a phd in applied mathematics, and still have to stop and ponder quite a bit during the last few parts. Also I would love more videos on group theory/Galois theory :)

[@soumyadipsarkar5078]

if you can put up some videos for group theory,then probably i would be the happiest person in the world.

[@mikikaboom9084]

I have no idea who dislikes videos like this. The amount of effort put into it is just tremendous.

[@Mathologer]

There are a LOT of really obnoxious cranks/trolls I have put in their place in no uncertain terms over the years:)

[@CynHicks]

Maybe their mom left their dad for a really smart, bald and sexy man when they were kids.

[@shreyan1362]

@@CynHicks 😆

[@govindasharman425]

These*

[@govindasharman425]

@@CynHicks It smells like Johnny Sins in here

[@pedrocleezz4315]

Do you really think im going to watch the whole 50 minutes? Because you are totally right

[@Mathologer]

:)

[@Sam_on_YouTube]

Nobody who watches Mathologer videos is put off by them being long. Just ask 3 Blue 1 Brown.

[@asktnegi5723]

@@Mathologer yes I do. Because I love to make experts correct when they do something wrong. But man this video just blew my head. But my wit and will to make experts correct is damn huge, so I will continue to work on these summation formulas even though It will take a Quadrillion year! BTW Congratulations to Mathologer for hitting 500K. You know I really don't get sick of watching your video till end. I enjoy it. And trust me I mean it from my heart. !!! LOVE YOU MATHOLOGER!!!

[@asktnegi5723]

One Day I am Gonna beat you for 50 mins sake!! ( I am just a student in grade 9 btw)

[@saulsavelis575]

u bet I am

[@leobidussi5039]

The last remarks hinting to asymptotic series had me on the verge of my chair while watching the video. I am a physics PhD student and your "masterclasses" are gold to me even if they have nothing to do with my research. The amount of care, precision, passion and entertainment that is present in your videos is outstanding; making such complex and rich subjects accessible to the amateurs and casuals as well as interesting and non trivial to the experts is a miracle made possible by your talent and commitment. I am not saying this to appear sappy, it is just that I believe that it is priceless the fact that I can have access to high-quality content like this for free. I just want to show appreciation for the creators that spend time and energy to put together a video like this one. Thank you very much and keep up the good work!

[@maximilianschmidt6708]

I completely agree with you! Thanks for your comment, a fellow physics PhD student :)

[@rafaeln2901]

ME GUSTARÍA DE VERDAD QUE EL MAESTRO QUE HABLA LO HABLE EN ESPAÑOL NO EN INGLES PORQUE NO LO SE , SE LO AGRADECERIA MUCHO GRACIAS DIOS TODOPODEROSO BENDIGA A TODOS LOS QUE HICIERON ESTE VIDEO DE MATEMATICAS.

[@krishnachoubey8648]

I agree. Thank you yt creators, and you, yes YOU! sir

[@umartinko]

So well put, I wish I could like your comment a hundred times. Thank you for taking the time to comment - I'm glad that it got the attention it deserved.

[@EssentialsOfMath]

I really appreciate the "insane" videos. A lot of math content on KZbin is watered down or repeated, so it's nice to see this stuff that I've never heard of.

[@jonathangrey6354]

Is there an option to like multiple times?

[@PC_Simo]

@@jonathangrey6354Create multiple accounts, and like with them all, I guess. 😅

[@rickzegooene]

Really appreciate how much effort is put into these videos. Very well structured and illustrated. KZbin really doesn't deserve this level of effort.

[@asktnegi5723]

People like him are giving me hope to study hard.

[@SledgerFromTDS.]

The truth is that I actually like KZbin for many reasons personally.

[@thek3nger]

Wow. 50 Minutes of Mathloger. It is like Christmas already!

[@Mathologer]

Actually, not Christmas but Halloween (< t-shirt :)

[@LarsHHoog]

I came to think of the computer science joke about Oct 31 = Dec 25 :)

[@Mathologer]

@@LarsHHoog I've actually got a t-shirt that features this joke. Pretty sure I wore that t-shirt in a previous video :)

[@LarsHHoog]

Where do you get all those lovely t-shirts from? By the way, this video was not as tough on the brain as numberphile's on which of TREE applied on Graham's number of the other way around is biggest. That was like putting your head in a vice between two black holes. @Mathologer

[@LarsHHoog]

@@Mathologer kzbin.info/www/bejne/ZomcdYyIgdKHhLs

[@bastiaanabcde]

Feedback: I've just finished a master's degree in mathematics, now starting with a PhD. Nevertheless, there are many things I'm learning from these videos. It's cool seeing recreational math videos reaching this level.

[@abdurrahmanlabib916]

I also want to study maths but the problem is there are no high sàlary jobs unless u do a phD :(

[@knudsrensen2486]

@@abdurrahmanlabib916 If you combine it with computer science there is a lot of jobs.

[@paulg444]

The first qualification of a PhD: love learning.

[@samm-r3u]

I was feeling like I'm dumb when I couldn't understand it and Im in 10th standard but....

[@keithjohnson6765]

@@abdurrahmanlabib916 'data science' field and a b.s. and you can do very well at least in USA... even better with a stats masters. 8) you don't have to have a PhD

[@rchas1023]

I graduated over 55 years ago, and am now retired. I've always had an amateur's interest in number theory, and am gradually relearning things I never really understood at the time. Thank you for the care you put into these videos.

[@lekhapratap1652]

Twenty minutes later I felt the power of Matrices.

[@sakshamtyagi6527]

@Deepak Velani जय श्री राम

[@quirtt]

@@sakshamtyagi6527 isse uska kya connection hai? Lmao

[@Randompersonon-f5u]

All KZbin:trees Mathologer: let's sum quadrillion of powers

[@chirayu_jain]

Who thought about numberphile here

[@xyz.ijk.]

Trees(g64) ... that's still pretty cool as well.

[@IshaaqNewton]

@@chirayu_jain Don't be everywhere, please.

[@TrimutiusToo]

Numberphile tree video had more powers than you can handle... But other than that i see what you mean...

[@rylaczero3740]

This video is g64 times impressive than the g64 video. I realised it was 50 minutes long only after I had watched it, felt more like 10 minutes long. This channel makes math so much more interesting.

[@unvergebeneid]

I get the feeling that Euler was a pretty smart guy.

[@davidrheault7896]

He was able to compute 6th powers mentally

[@alexandermizzi1095]

Your profile picture seems familiar

[@IslandCave]

He was so smart, he had a 1 Up on everyone else!

[@reznovvazileski3193]

The cops once asked him for his identification. It appeared he had quite a complex identity, but it turned out to be real.

[@mariomario-ih6mn]

Lol I subscribed

[@austinyun]

27:46 when he paused, I was already like 90% sure he was going to say Euler. If you're ever on a quiz show and one of the questions is "Who first proved X?" if you just guess Euler you're probably good.

[@lhibbeler]

I hold a PhD in mechanical engineering. I had several graduate level math courses along the way, and I continue to self study on topics that I didn't have time for in school. I vote for going deeper / more intense. Every video of yours that I have watched has been quite enjoyable. I like how you start with very elementary ideas and before I know it, 50 minutes has passed and I have a new tool in my toolbelt.

[@accuset]

I'm an undergraduate ME and I really enjoy seeing the stuff I used to hate (mainly Cal 2 topics like infinite series) turn into this ugly puzzle yielding cool results. Even if it isn't immediately practical, understanding e^(i pi) + 1 = 0 by itself is very rewarding. Or watching someone use a matrix as an exponent, which I've never even heard of before. It feels like humanity is progressing somehow lol.

[@PC_Simo]

@@accuset ”It feels like humanity is progressing somehow lol.” I sure am hopeful.

[@Anistuffs]

"Don't worry. Be happy. And let's leave the demons for later." - Life advice from Mathologer, 2019

[@Your_choise]

The demons were saved for 2020

[@Mechinbro]

I rarely ever comment but my appreciation for these master classes merits a comment here. Other channels either prove easy results that look pretty in animation or mention crazy results without giving proofs. What I like about these videos is that they actually take the time to prove those crazy results. I would love to see any videos you may post on Galois Theory or any other weird topic only math insiders get to see and that other channels say "so and so proved this 100 years ago but the proof is really hard"

[@thatonemailbox]

Chapters: Chapter 1 - (Little Gauss) [4:25] Chapter 2 - Proof without words [10:04] Chapter 3 - ??? [???] Chapter 4 -Pascal and his triangle [19:50] Chapter 5 - The Bernoulli Numbers [23:23] Chapter 6 - Infinite power sums and integer values of zeta [30:06] Chapter 7 - The Euler-Maclaurin formula [32:42] Chapter 8 - Euler-Maclaurin jump starts The Basel Problem [42:14] Keep in mind that this video is meant to be watched from start to finish-don’t skip ahead just because you know already know what he’s talking about, it’ll only confuse you. I’m only leaving these timestamps for anyone wanting to keep track of where they are in the video.

[@michaelguenot6177]

3:55 omg I just realized that’s a pumpkin pi.

[@Mathologer]

:) Just getting ready for Halloween (big thing in our family :)

[@theunknown4834]

@@Mathologer why not do the trees thingy everyone is doing Actually don't, I like your unique videos

[@anteconfig5391]

@@theunknown4834 Actually I think this is kinda like a tree video. If you start off with on object in the first row and force it to have a maximum of two children and take those children put them into the second row and keep going forcing each child to give birth to 2 more entities and place them into the next row you would be making a binary tree. The number of entities for the nth row of this tree is equal to the sum of the nth row in pascals triangle. and pascals triangle was used in this video to do these power sums. In some sense it is another tree video. PS: In my mind everything is the same thing, which would mean that all videos on the internet are tree videos so this comment might mean a-whole-lot and a bunch of nothing at the same time.

[@inyobill]

Yah, I was immediately: "I need one."

[@timbeaton5045]

Or wear two shirts for a Pumpkin Tau. Oh. That doesn't make any sense... I'll get my coat........

[@TheOneMaddin]

This was the best of your videos. I am a PhD student of math, and today I learned something new which I have to look into much more. Definitely more of this please. Complicated math is my bread and butter and you are one of the few youtubers who dares to go into the details. The length of the video was absolutely appropriate.

[@AdhiNarayananYR]

Hey, I’m from India. I’m an engineer/ data scientist. Currently working with financial models for derivatives. This video, was it really 50 mins? It got over pretty quickly. I felt like the video was just getting started as it ended. Fantastic stuff. I think i was able to follow till the end. Can’t wait for the next one in this series.

[@PC_Simo]

Hey, I’m an India-fan from Finland. India has a great culture, *_DELICIOUS_* food, and super-cool people. Love to India from Finland 🇫🇮❤️🇮🇳.

[@unnoticedspacegoat8537]

And as always the first person who prove ... Leonhard euler

[@davidrheault7896]

The master of us all hehe

[@JCOpUntukIndonesia]

Your explanation did work on me prof. Burkard. My background is engineering, and I played a lot with Optimization, Partial Differential Equation and Numerical Methods. The 50 mins long does not bother me. I believe 50 mins is just right to sums up millenia of math discovery. The moment you said 10 chapters really sparks my joy, and finishing the whole video is really satisfying. This is indeed a Masterclass professor. Can't wait for the next. Thank you very much!

[@akanegally]

Your work is simply amazing. I like how the difficulty rises. I always have something to pause and ponder. As usual I will re-watch it a couple of times to try to understand the last part.

[@diegofernandessantiagoribe6996]

Thank you so much, you helped me before and is helping again! Your work is fantastic

[@modestorosado1338]

I'm always looking forward to these monstrously long videos from the many Math-oriented content creators on KZbin. In particular, I feel the animated approach in these videos really helps to visualize the most intricate arguments. I should also mention that your work is very enjoyable to mathematically inclined people, mathematicians and non-mathematicians alike. And even if one has a deep background in Mathematics, there is always something new to learn.

[@admiralbananas]

Truly awesome video. I'm a bachelor's math student and never felt lost throughout the entire video, and this is my first introduction to Bernoulli numbers. I will need to watch the last 15 minutes a couple more times to feel like I have a more profound understanding I think, but I will certainly do so. This video really left me itching to learn more about these infinite sums and definitely more about Bernoulli numbers. I'm looking forward to the follow-up video!

[@KSignalEingang]

Since I made it to the end of this, and since you asked nicely: I'm an enthusiastic amateur, never took much beyond AP calculus formally but I read & study the subject pretty broadly as a hobby. I was cruising along smoothly here right up until about chapter 4 or so, after which there was a lot of pausing, rewinding, workings-out on paper or Python, and a couple side trips into Wikipedia & Wolfram. The Pascal/Bernoulli thing was mind-blowing to see in action. It definitely made sense on the surface, though as it got deeper i started to feel like it was going in circles? Something like, "We can easily derive S using B. But how do we get B? Well by deriving it from S of course." At least that was my initial impression. Really enjoyed it even if it did start to outpace me towards the end (or more accurately, *because* it did). It'll be a few more viewings before everything clicks for sure, but I'm looking forward to the challenge!

[@tomkerruish2982]

28:12 The error is in the final term for the sum of ninth powers, which should be -3/20 nn.

[@Mathologer]

Yes, full marks :)

[@firstussecondus1088]

I found the same term by simply adding up all coefficients for each line; which should equal 1when n=1. I noticed that the 9th added up to 1 and 1/15th. By adding that extra 1/15th to 1/12th the result was 3/20th.

[@performancetesting1]

This channel gets me so pumped on getting more math skills. Thank you, Mathologer!

[@vsandc]

Excellent video. Best 100 minutes in the past 24 hours. Thank you.

[@mkb6418]

I did this in class in first semester. I was a kind of dizzy, thank you for helping me understanding and appreciating this after so many years.

[@Kram1032]

I love it! What a great video! Your deep dives into maths are amazing and I can't imagine just how much editing goes into all these presentations.

[@lowercaserho]

My background in maths: two years of an undegraduate degree in theoretical physics before dropping out due to ill-health, then haven't really done any formal maths for the ~15 years since. My experience with this video: I was following everything fine up to the Bernoulli numbers; from that point on I was following enough of what was happening to still find it interesting and enjoyable to watch, but I'd have to go back and rewatch -- probably with multiple pauses -- if I wanted to be able to say that I truly understood it all.

[@caloz.3656]

Everytime I watch a long Mathologer vid it feels like I'm watching a movie. Thank you!

[@craigwillford5113]

Your requested feedback: my B.A. was in mathematics. My favorite class was number theory. But after undergraduate, I went to law school and became a lawyer. Mathematics became a hobby. I am stunned at the quality of your presentation! The çolored boxes, the floating powers and variables, etc. make the ideas being presented so very çlear. Truly excellent work. Thank you for all your time.

[@Mathologer]

Comments like this make my day :)

[@williamfloyd9024]

Bachelors in math checking in: The video worked very well! I feel like I have a good high level grasp of the concepts now. As far as feedback, the pacing was excellent and I think you should continue omitting some of the “dryer” parts of these proofs. This keeps the pacing very good and all the content interesting. I recognize that they’re very important, but I never much cared for proofs of things such as convergence or uniqueness. I think acknowledging that some steps are omitted and providing details in the description is a perfect way to handle these necessary evils of proofs.

[@gman064]

I love watching your videos even though this is way out of my field. You make math very interesting and dare I say it? Fun. Thanks for the hard work.

[@sterlingveil]

I already had a python interpreter open, so I did Bernoulli's sum in about 20 seconds! 😝

[@Mathologer]

:)

[@brucefrizzell4221]

Did you use the python Bernoulli library ?

[@herrfz]

>>> sum([x**10 for x in range(1, 1001)])

[@OlafDoschke]

@@herrfz This works in split seconds, yes. Thanks to arbitrary long integers.

[@ScienceSupersedesFaith]

Dammit... I wanted to be the one to write this comment first. However, 20 seconds is very long for this task if the interpreter is already open. Typing the code should last less than 10 seconds and the actual calculation takes only a fraction of a sec. ;-)

[@squidgyfishy]

Mathematics BSc here who rarely uses the technical bits in my day to day, but you keep my love of the subject alive! Very very grateful to you for all the time and effort you put into making these complex topics accessible and bringing me some joy whilst you're at it!

[@Hampardo]

Physics graduate student here. I loved your video. Everything was crystal clear. I'd love to see more of this Euler McLaurin creature.

[@tazking93]

These videos are categorically beautiful. With such a transparent love of mathematics, and talent for education, this truly was a gift to the world. As per your request, I’ve a BS in physics here, and while everything seemed reasonably intuitive as you explained it, I’ll need to spend a few hours going over it carefully to deal with the technicalities. I’ll wait with bated breath for your next video.

[@groupoids]

Re: the end questions, undergrad computer science, and would be fantastic to see more of the "insane stuff" :)

[@GreenMeansGOF]

I have a bachelor’s degree in mathematics. I was able to follow everything except for the formula at 40:45. Im sure that requires more thought/explanation but overall, great video.

[@Macieks300]

I thought about it and it's not that hard to derive. I took the difference of the two sides of the equality and substituted the formula for f(x)=c_0+x*c_1+... for all the derivatives and the antiderivative. Then I grouped it all by the c_k coefficients and got to the equality imgur.com/cCDdHCt. Now it looks like some kind of known equality about the Bernoulli numbers but I don't know that much about them.

[@alonamaloh]

@@Macieks300 I think it's just applying the formula to f(2)+f(3)+...+f(n), and then adding f(1). If you managed to prove that's equivalent to that identity about the Bernoulli numbers, then you just proved it. :)

[@JorgetePanete]

I'm*

[@nicholaspotter9925]

Exact same situation here.

[@columbus8myhw]

Add and subtract f(1) in a clever way

[@Muhammed_English314]

In 2019 I was studying Chemistry and one exercise caught my attention , it was something like " mix two of the following materials to get the result" and I started thinking how many possible ways we can mix these materials together and notice that: 1- order doesn't matter , 2- you can't mix the material with itself (i.e. you can't repeat the same material). After playing around with this idea for quite a long time I noticed some patterns and found relations between P2 and p3(Pk is the number of possible pairs of k) then I stoped. A few days ago I took on the challenge again but there was that thing that stoped me from finding the formulas for Pk(N) = the number of possible pairs of k from N elements , that thing was S2 and I took another approach of finding the formula : 1+2^2+3^2+...+x^2 = 1+2+2 +3+3+3 +...+ x+x+..+x(x times) then rearrange it to : 1+2+3+...+x 2+3+...+x 3+...+x . . . x basically : S1(1 to x) S1(2 to x) . . . S1(x to x) i.e. The sum from j=1 to x of the sum from n=j to x of n = The sum from j=1 to x of ( S1(x) - S1(j-1) ) = xS1 - ½S2 + ½S1 3S2=(2x+1)S1 S2=x(x+1)(2x+1)/6 after that I found the general formula for my original question Pk(N)=(1/k!)Π(j=0 to k-1)(N-j) Then I started looking for other powers and found formulas up to S5 using the same method I used above and I was wondering if there is any pattern here. After watching this video which I kept ignoring since I wanted to do it myself , the answer is yes but actually no , there isn't any "easy" formula to find the sum of Sk(n) but I now know that there is something that nobody has done before (the odd power monster). If you made it till here then I hope you understood my symbols also these kinds of long videos are awesome for those who really like Math and want to go deep into it.

[@ganesansubramaniyam5538]

This man had made a 50 minutes long video about this interesting topics. What an effort ! Hat's off to you sir. Hoping that you make a Guinness World records on longest video about mathematics. Love you sir. Proud to be your subscriber

[@ericbromberg6022]

Loved the long and in depth video! I'm still in high school and really had to fight through a lot of this, but absolutely loved learning about the more technical side of mathematics! More like this!!!

[@jacksonstarky8288]

50 minutes that didn't feel like 50 minutes. I was ready for you to get into the zeta function and gamma (both the function and the Euler-Mascheroni constant). Re: your request at the end: I'm coming up on 30 years out of high school and I haven't taken a formal mathematics course other than introductory statistics since 1992. I got into recreational mathematics thanks to the late Martin Gardner and a few other writers, and then I found channels like yours and 3blue1brown on KZbin a few years ago. Lately I've been pondering the Riemann Hypothesis and the Collatz Conjecture, and I'm currently re-reading Julian Havil's book Gamma, about the Euler-Mascheroni constant. My instinct is that it's transcendental, but I'd love for somebody to prove it. I'm really looking forward to seeing what you have to say about gamma and the zeta function... and take as long as you like on those topics! Edit: Yes, I've seen your previous videos on Riemann and related topics. Also, something I remember discovering while playing with numbers on my own in high school algebra class came to mind. I remember finding that the perfect squares had differences of consecutive odd numbers, and then when I took this to higher powers I found the factorial of the exponent at the bottom of the formula, but I was never able to complete the formula for the full general case. I still have my original spreadsheet file from 1990 with my calculations.

[@snillie]

Huh, I just left high school and I think I recently discovered the same thing you mentioned :D I'm pretty sure I got the idea after finding out taking the sum of the first k odd numbers gives you k^2, and I eventually got the idea to take the sequence of the n-th powers, take the difference of each consecutive term to generate a new sequence, then if you do the same thing to THAT sequence and keep repeating that process, you seem to eventually run into a constant sequence where each term is equal to n factorial. I actually explored this idea a lot and eventually came up with an explanation for why n! shows up there (along with some other interesting things), but sadly I never wrote it up in any coherent form. Though now I kinda wanna try rediscovering everything I found out and actually write it up somewhere ^^ It's really cool stuff!

[@COZYTW]

Regarding the Collatz conjecture, I've been going on fumes as well. For the most part, I needed to find a number that satisfies the following formula: 3n + 1 = 2k (2n) where k >= 1. However, the only number that seems to satisfy this inequality so far appears to be 1, which makes k = 2. All other numbers seem to fall below this constant. Well, for the most part if we really wanted to breakthrough / contradict this particular conjecture, my guess is that algebraic methods only serve to cement the fact that all numbers trickle to 1 and they won't get trapped in loops like 1 itself does. Or, that my current formula's too simplistic. Polynomials stuck at the power of 1 usually only have 1 solution, as is the case with power of 2 = 2 potential roots, powers of 3 having 3 potential roots and so on. Trying to let it catch isn't quite as easy, and I've been trying to work out possible constants/coefficients that might affect the outcome, similar to how some of the terms in the video have a coefficient of 0.

[@jacksonstarky8288]

@@COZYTW I found basically the same thing you did; finding n=1 and k=2, and nothing that worked at any higher numbers. The Numberphile videos on the Collatz Conjecture were my introduction to it. First, this one: kzbin.info/www/bejne/a56poYl6pbCLbZI Second, this one: kzbin.info/www/bejne/gqKuoZ6ah7d_kNk The first video shows some interesting behaviour of the loops when the Collatz equation is modified, suggesting that if all numbers do in fact reduce eventually to 1, it's not true for all variations or similar formulae. The second video reveals a very "organic" pattern... and the first video suggests that this problem is unsolvable, which of course is why it and the Riemann Hypothesis have caught my attention. The Millennium Prize for Riemann is definitely a nice bonus, but my real motivation is the beauty of the mathematics, especially as captured by 3blue1brown.

[@erikpedersen9400]

I don't understand that notation: 3n + 1 = 2k (2n), what's the (2n) mean?

[@jacksonstarky8288]

@@erikpedersen9400 Here's the Wikipedia summary of the Collatz Conjecture: "The Collatz conjecture is a conjecture in mathematics that concerns a sequence defined as follows: start with any positive integer n. Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that no matter what value of n, the sequence will always reach 1." So... the 2k (2n) is the even term (hence always a multiple of 2), and the (2n) is in parentheses just to show that the 2k is a reference to a term in the same sequence. But to say that 3n+1=2n gives us a result with no integer solutions except -1, so we have to use a dummy variable... which, I agree, can make things confusing. I hope this clears things up.

[@aliberro]

The best of the best channels ❤️ with high qualitt content, we love ur animation and appreciate every second u spend in preparing them

[@beanhwak]

I am a retired math teacher but still enjoy your amazing videos. I extract some of your contents and share them with students who are training seriously for math competitions. Thank you, Math from the Heart!

[@Socrates...]

I love this guy, he shows me beauty of the mind. Have a good New Year. I hope you make many more videos like this. Thank you.

[@marccaballer]

16:50 Shouldn’t it be (x-1)º = 1?

[@Mathologer]

Yes, it should :( Damn typos.

[@Mathologer]

@Subscribe to Me for Blessings 50 minutes, I bet it's typos :(

[@7654jake]

not when x = 1

[@xyz.ijk.]

@@7654jake 0^0=1, no?

[@Kartik-yi5ki]

@@Mathologer No probs, even bernouilli did it

[@brucedickson6019]

Quite wonderfully lucid! I particularly enjoyed the 2D & 3D animations - they lend a new perspective to otherwise very dry algebra. Since you ask, I'm a physicist with advanced degrees in Cryptography and Applied Math.

[@marchenwald4666]

1:51 Challenge accepted ! (1/11) * 1000^11 + (1/2) * 1000^10 + (5/6) * 1000^9 - 1000^7 + 1000^5-(1/2) * 1000^3 + (5/66) * 1000 = 91409924241424243424241924242500 Pretty faster that way !

[@kasugaryuichi9767]

What is this magic?

[@AugustusOakstar]

I loved every second of this video. Your visuals take me by the hand and lead me through the process very smoothly. I have always loved mathematics, Dad had a 4 volume set "Men of Mathematics" by Bell, it was my secret weapon in the '60s when I was in grade school. It made me appear to be so far ahead of my classmates..... Then I realized that math is the ultimate toolbox. I used it all: from A to z, statistics, 3 semesters of Calculus, Vectors, Matrices, etc. I became an Electrical engineer, but along the way I was published in the Journal of Surgical Research, because in high School I worked at the Med School analyzing data for grad students. Then the computer appeared and of course I could use algorithms with great facility allowing the automation of my analysis. Please keep going, you are never boring. With greatest Respect, Wm J Ackley😸

[@glumbortango7182]

Mathologer, I write this comment to confess that over the summer I've independently invented the Euler-Maclaurin Sum Formula out of spite towards the many Quora forums saying there's no solution for the continuous summation of a function because it "isn't defined for non-whole numbers" as if the best you can do for an answer is grab a sharpie and connect the dots by hand. How I did it was part knowledge, starting with the Taylor Series and some known sums and going from there. Part of it was also blind luck near the end where I stumbled into the Bernoulli numbers looking at the Wikipedia page for the Zeta Function, and noticed the exact sequence in my notes. Even after it was all done and I found the formula, I didn't know the name until this video. Seeing my notes over the past few months laid out in a video so cleanly has been a very special experience, I'll say that much. Besides all of that I think I'm going to start watching this channel a bit more closely, if not for the unflinching look into mathematics then to at least save a few months of work in the future. Thanks for this video especially, it's more than valuable to anyone looking into the questions I started with.

[@JRaiKetchum134]

Feedback: Science Teacher with BS in Molecular Biology and Chemistry Highest Math - Multivariable Calc Overall, I was able to follow through to the end fairly easily. You always do a great job in explaining these higher-level concepts by stepping us up from the basics. The part I got a little bit confused on was where the factorials came from when deriving the Euler-Maclaurin Formula. Then, I went back and watched that part again and figured it out. 50 mins is nothing when you’re deeply immersed in the video!

[@soumyadipsarkar5078]

This video made my day,i was waiting eagerly for a mathologer video

[@TheNameOfJesus]

The one thing I was able to remember from this video was that historical mathematicians said "nn" instead of "n^2". They only started using digits for powers starting with "n^3."

[@ajokaefi]

Since you asked for feedback, here it is: Both me and my son watch and appreciate your channel from Cabo Verde (portuguese speaking), ... the carefull and visual explanations are unmatched online!

[@matthewrobertmeadows]

I really love how you make complex topics easier to conceptualize.The videos are quite entertaining and i love watching them

[@frolnek]

I WATCHED ALL OF IT ! ! You are "AMAZING", I watch as many as i can. I am 78 years old. I only got to Algebra 2 in my youth. I wish you had been around 60 years ago. Don't be fooled ... I do not understand most of it, but you are never too old to lean something. 6 STAR INFO

[@karangupta1825]

Leonardo DaVinci said "Learning never exhausts the mind".

[@chadx8269]

Keep it up!

[@NT-nw9ek]

That was great! For the question at the end: I'm in my third year of a B.S. in EE, and the part that had me scratching my head for the longest was the rewriting of the formula around 40:43. Would love to see you cover the foundations of Galois Theory!

[@tonecukon91]

Feedback: Mechanical engineering student with interest in math . Think I got almost everything. Will try to do the bits you left for the viewer. Thank you for awesome videos like this. :D

[@UltraViresAdInfinitum]

Thank you Mathologer! I am a high school math teacher and these videos really keep my spirits up and gets me pumped to teach!

[@rgarciaf071]

The best used 50 minutes of my weekend, love your content!

[@bemusedindian8571]

Galois theory... I cannot wait to see how you approach it. Even the chapter 7 and 8 here exceeded my expectations. These are topics where the teacher learns more each time they try to teach someone.

[@atharvas4399]

this should be awarded the best animation of the year award.

[@TrimutiusToo]

This reminded me of my Calculus classes back in University... I have master degree in physics, which required a lot of calculus class... The only thing that this introduced to me (or maybe i have just already forgotten) is the source of Bernoulli numbers... But approximation you showed in the end was not new to me... I wasn't required to know it for exam but we were introduced to it in passing... I just like your explanations that is why i was watching... Not for new stuff but for new insights into stuff i more or less already known so far... A different angle you take to show known mathematical results is super awesome...

[@charlieb8735]

The passion for this material and command of it are the type of thing that makes me watch Mathologer videos whenever they come up. I don’t have much besides some basic college math as far as formal education but I have a great love for math that I had up until high school. What is special about this channel is that it gives full explanations and respects that I’m here to actually understand. Some videos I don’t get everything, but as someone who had been independently relearning and now is expanding my mathematics as much as possible, the insights that can come from any concepts connect and recontextualize others. Coming back and rewatching videos then actually getting the concept that I just wasn’t able to or finding a new obstacle to study thats within reach. In short I hope that this channel always continues digging into this type of material. Very few else do and of those, I don’t know of one with the quality of explanation and depth I find here.

[@anantrajagopal1338]

Ref: Question at the end Prof. Burkard, sir, I am a third-year undergraduate computer science student from India who loves mathematics and was a National level participant in IMO-2016. Your videos have helped me continue my interest in mathematics and I must say, many of the things you put forward in the simple ways (e.g. using the Pascal triangle to calculate Sum[n] formulae, not only help me and others get new approaches to problems but also come in handy in problems in computer science. I truly appreciate the level of work and detail you put into each video and hope to see more of the same in the future. Thanks.

[@alexandermcmiller6175]

I am studying math in university right now, and I am not put off by such a long video. I had to split it up into chunks for when I had time, but it was intriguing, so obviously I watched it all the way through, and if you continue to produce videos like this, I would watch it even if it was 2 hours.

[@samarthsai9530]

This was incredibly amazing....so much fun... I am a high school student 11th grade and we were studying sequence and series of course not of this level but this was insane. Thanks.

[@MYPASSIONMYLIFESTYLE]

2:09 it's pretty easy to see the symmetry in the number. When I tried to calculate that number by hand by first adding the positive terms of the summation formula we get a number / 132 when you take the LCM and we get a recurring decimal up to some point. this is why we get the numbers repeating in pairs of 3.

[@Primalmoon]

Thank you for the animation at 11:21 . A decade ago I was tasked in my combinatorics class to derive that formula for 1^2+...+n^2 and I was never able to get the right arrangement of stepped pyramids in my head for it to make sense, and I couldn't figure out how to draw it on a paper without getting lost. I knew I was on the generally right track but eventually had to look it up online without getting a good intuitive sense for the visual / combinatorial proof. And that has still bugged me as a combinatorial proof that I couldn't figure out for myself. Now I see I was working from a harder starting point in that I was working up from the proof that 1+...+n is not half of a n*(n+1) rectangle, but from an (n+1)^2 square that you subtract the diagonal from and then divide by two. Trying to increase the dimensions of that proof up to 3d still gets you interlocking pyramids in an (n+1)^3 cube, but the error terms are complicated to think about when they're slices of a cube following the surfaces of 3 square pyramids.

[@premmistry7]

I'm from India and I have just completed my high school and I love to watch your videos, They are just mind blowing and now I love your channel, at first I used to watch numberphile videos and found them interesting, but your videos are now more interesting and intuitive unlike numberphile! Hats off sir, please keep making videos and everyone will love it!

[@garfieldjhones7887]

Awesome vídeo! I like the increasing difficulty through the lesson, It keeps the video accessible and still challenging for everyone. I'm a nerdy high School student from Brazil xD I followed everything until the integrals after 38:00

[@ducktectivewhitewings9276]

48:47 my feedback: i'm a undergrad engineer, and loved the video. The explanation was very good, and very gradual. please make more :)

[@julianfess98]

I did know the Euler McLaurin from Wigner-Weisskopf Theory in Quantum Optics, but for me as a Physicist ist was just a mathematical tool to solve a hard integral Thanks to Mathologer I now know the true beauty and significance of this formular To answer the question, all of it worked for me, but I'm studying Physics so my background in math isn't that bad Grüße aus Deutschland

[@discussionfortherightway.9725]

Grüße auch von mir.

[@deathclockthegreat2882]

Amazing video. I specially loved the animations for the pyramids showing how to calculate the sums. To me, the part that got me awestruck was at around 17:13, where you use (x-1)^5 to get S_4. I found it very simple and elegant. It seemed to me that right at 18:35 you would derive a generalized recursion formula for the S_n which I felt was an incredible result. For anyone interested in the recursion formula. Following the exact procedure (beggining at 17:13) but instead of 5 putting k+1, the formula you would get is: S_k=(1/(k+1)) * [n^(k+1) + Sum_{i=0}^{k-1} bin(k+1,i) (-1)^(1+k-i) S_i ] bin(k+1,i)=(k+1) is meant to be the binomial coefficient ( i ) Really can´t overstate how much I enjoyed the video. PS: I´m a fourth year math`s student.

[@StuartBrorson]

Although this video is 5 years old, I wanted to say thank you, and please make more like it at this level. I am always pleased to find high-level content on the web, and I really enjoy and appreciate your math videos. They're great!

[@holomorphicguy]

Finally!!! I've been waiting for it.

[@mitchkovacs1396]

Loved the video! Definitely really easy to understand compared to other Master Class videos. I think it helped that it was very interesting so taking it one step at a time didn't cause me to lose interest

[@DoReMeDesign]

I love this, I was reading some old English textbooks from the 50s and was impressed by their derivation of power sums for high powers using a recursive model. And you've generalised it!

[@jayprich]

Loved this. I came here for the graphics but stayed for the references - so many great jumping off points and further readings! I'm now 55y old but did once do a maths degree, your approachable style has me interested in exploring fields of the subject I was previously too daunted to even try. I am still daunted and rely on you chase away those demon tricky details!

[@srila6893]

More than a decade ago, I struggled through trying understanding the links between these powers. Your video completed my quest in about 1/2 hr. Thanks for this great insight. I have been watching a few of your derivations recently. They are really good. Love it. Cheers

[@bhaskarbagchi1643]

This was amazing. Never expected to see a popular exposition of the Euler-Maclaurin sum formula. And you have done a great job. I am a retired professor of mathematics, specializing in finite geometry and combinatorial topology. I have a question. Doesn't Ramanujan have a "Pi-based" Formula for Zeta of 3, even if it's not the kind you had in mind when you challenged your listeners to find one. If my memory serves me right, Ramanujan's formula involves a crazy ( rapidly converging) infinite series, or maybe two of them. My regards.

[@dougr.2398]

More details, please?

[@Koisheep]

This is the reason why I mayored in math: to watch Mathologer master classes from start to finish

[@MarkGerads]

This is great, and I am looking forward to that hour-long Euler-Maclaurin sequel.

[@dougr.2398]

Check back occasionally and please post if you find one?

[@thesparksplug]

I fell in love. ❤️💕. I did my degree in Maths in 1997 and you, Mathologer, just keeps telling me what I was doing all those years. I love it. So sorry I didn’t have these tutorials back then!

[@sxz452]

This is ABSOLUTELY one of my FAVORITE KZbin videos about math.

[@arberithaqi]

I didn't know that you were german. Only this video showed me, how perfect you pronounced the names of the mathematicians. Gutes Video wie immer und Gruss aus der Schweiz ^^

[@Mathologer]

Yes, I grew up in Germany. I moved to Australia about 25 years ago :)

[@stabgan]

I wanted to see a simple conclusion recap at the end of the euler formula. I'm an IT engineer, I understood every mathematical aspect of it except it was a little overwhelming to describe to someone else just what I saw.

[@andlabs]

31:38 For a long time I was hoping the pi^2/6 identity and its relatives could be held up as a counterpoint to "everything is easier with tau" zealots. Then, wanting to know if there was a general formula for the multiples of the powers of pi in those identities (whose numerators and denominators have OEIS sequences, btw), I learned about Euler's general form for those sums and how it contains 2*pi as a unit. 😐

[@dlevi67]

Look no further than the gamma function...

[@shahriarhossain2317]

Mathologer is my favourite KZbin channel now. Learning deep stuffs on math is not the most important thing here. Rather the most important thing is that it increases our love for math and encourages us to gain more knowledge about math. It also encourages me to choose mathematics as my subject when I will be a university student.

[@rzipstein]

Recently discovered your channel and love it. The topics you choose are fascinating and the presentation is clear and entertaining. I am a 51 year-old retired photographer working towards a master's in mathematics and a second career in teaching. Keep posting great videos!

[@YellowBunny]

I'm still waiting for a follow up video on rooty expressions by the way :P But this was also a very interesting topic and I was able to follow along quite well for the most part. Only during chapters 7 and 8 there were some calculation where I just believed you to be right because I could not check them in my head.

[@3Jeroen3]

Phew! Got through it but need to rewatch how those crazy calculations in a pre PC era where done. My background is in computer science.

I'm now like Oliver Twist: "Please sir, I'd like some more!" :D Feedback: I could pretty much follow it all to the end (though I'd prefer some of the skipped and glossed over steps be included, because I had to pause a few times). I'm not a mathematician (but I have a master's degree in theoretical physics).

[@mattialuca100]

I'm on youtube since 2008 and this is one of the FEW 50min long videos I watched without skipping any second. Really really interesting and well explained. Keep it up!

[@frogamelanchier]

You asked for comments from those who made it all the way to the end, so here goes. I have a PhD in math, and I followed the entire thing and learned from it. Kudos for doing such a great job of making these topics available to a wider audience. Fantastic animations.