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

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.

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 :)

Enjoy :)

[pedrocleezz]

Do you really think im going to watch the whole 50 minutes?


Because you are totally right

[mikikaboom]

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

[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 YouTube 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 :)

[leobidussi]

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!

[EssentialsOfMath]

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

[rickzegooene]

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

[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.

[theknger]

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

[rchas]

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.

[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.

[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"

[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.

[Randompersonon-fu]

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

[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.

[Anistuffs]

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

[unvergebeneid]

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

[unnoticedspacegoat]

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

[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!

[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.

[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.