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

Excellent

Here is a recursion for the sums of powers: If f_k(n) = \sum_{i=1}^n i^k then (k+1)f_{k}(n)
=(n+1)^{k+1}-1-\sum_{j=0}^{k-1} \binom{k+1}{j}f_j(n).

Absolutely LOVED this video! As a University Student, my background terminates at only basic linear algebra + calculus, set theory, and the basic maths before it. However, I always greatly enjoy your videos regardless of how difficult is sometimes is to make sense of them!

What A amazing masterpiece :)

But you're all SO SMART,,👍😂

I'm George O'Shea. Here's what I found last. Did you or anyone else already know that..
45^2=2025.. is also equal to 9^3+8^3+7^3+6^3+5^3+4^3+3^3+2^3+1^3?!?!?!
And all those people who stole my inventions, the Artificial wave in Fresno, the cyclocopter, the toroidal propeller which I said I'd make look like a toroidal Celtic triskle. I said give that idea to mit. But I haven't been paid a penny and allot has been made.
It was all on a recording that was sent to the American embassy in Ireland.
I gave them allot of other inventions too and business investment ideas
I have more inventions that will never be made or shared if I'm not paid what I'm rightfully and lawfully owed. Homeless in Malibu California. What's right is right.
Have some actual integrity people ⚖️

Feedback?: I made it to the end, it wasn't that hard to grasp honestly (in year 10 accelerating year 11 maths methods) but that's probably just me. Although I didn't get the matrix stuff (I know it exists but I never bothered to learn how to use them). More "insane videos"? :D

Yes I know this video is 3 years old

I'm a retired mechanical engineer, and I have been working on finding closed forms for the odd integer zeta values. I purchased Mathematica several years ago, and have made some fantastic discoveries. One of my formulae will create rational number approximations to zeta 3 to varying degrees of accuracy. I think zeta 3 has been calculated to about 1.2 trillion digits.
I'd gladly post some of these approximations if anyone is interested. My work hints that closed forms to all the odd zeta values exist.

I liked the formula for the sum 1+...+n = n(n+1)/2 by considering a triangle plus the error near the diagonal. You can do the same thing with the sum of squares. The error term now involves counting the (triangle X interval) parts as two times the previous triangular number, plus n times the complement of a tiny pyramid in a unit cube along the diagonal, which of course has volume 1-1/3=2/3. So the whole formula is 1/3n^3+ 2*(n-1)*n/2*(1/2) + n*(2/3), which can be seen to equal the usual formula.

The sum of payments made to a pensioner on an annual or on a monthly basis form two approximations of an integral - the first difference lease to the so called woolhouses approximation to a monthly given annual annuity. The subsequent ones can improve the monthly approximation if you have enough knowledge of the mortality shape

Does the video on more Euler McLaurin formula currently exists?

I'm kicking myself for not pursuing a maths degree.

my background in maths well the highest I got was MSc by course work, but I quit when I could not cope with the continual low level sexual and religious harassment from my supervisor, she was one weird woman

hardly anyone outside Mathematics knows lots of things

that is exciting

you used all the calculus i forgot

Hey Professor! These are also Seki Takazu numbers

I am pretty sure that Euler-MacLaurin was used by Richard Feynman to compute thousands (?) of path integrals or components of his bubble diagrams. I once saw a page of his notes in a discussion with a solid resemblance of estimating an integral by a finite sum….. perhaps in a film

Could 3 to 10 or such n to 10 terms be represented nearby sum of squares or the sum of any set then n2+n/2 would be a correct interpretation.

At school, this ( functions) was presented as a slight of hand, wherein the student must guess how it is done. Now I know, 55 years later

Amazing video!!! Thank you so much. I really enjoyed watching it 🤤

I wish my number theory professor was 1/pi as clear and illustrative as this video is when he covered Euler-Maclaurin. Would've made my life so much easier. Really great video and appreciate your effort!

made it to the end,
I'm a highschool student and goddamn everything worked for me,
thank you

Go one by one.

mmmmm..... step-triangle, im stuck

Obsolutely fantastic please carry on this activities. Really interesting. Thank you

Why did you took x-1 in the first place for S4

Would be great if you would curse more. Personally, I really appreciate those existential crisis emotions of yours. Very relatable.
My background: tall, bald, wear black t-shirt, 31 y.o.

Thanks!

Can you make a video of ramunanjan sum of hypertrigo

Watched it again. wondering of any (more) connections to quantum mechanics. I would like you to through in hermitian matrices into the mix and whether normalizing a wave function in quantum mechanics has any links to the Euler-Maclaurin formula. What are the connections to uncertainty or exclusion seen in physical systems such as QED and QCD. In answer to question at the end of the video, I have a BSc education with a long life of continual learning.

Please, write this thing in a book. It is the best method to keep it. We need this...

The best part was Euler-Maclaurin formula

These type of videos are amazing after watching thesethere is so much stuff in the video on which you can learn a ton about.

Als ich noch aufs Gymnasium ging habe ich es geliebt Formeln für die Potenzsummen aufzustellen (mittels Differenzenschemata). Es war mir jedoch bis zu diesem Video nie bewusst dass man insbesondere die Summe der Quadrate so wunderschön veranschaulichen kann. Das werde ich meinen Kindern mit Legosteinen versuchen beizubringen. Und die allgemeinen Formeln über die "Pascal-Determinante" ist natürlich der Knaller, endlich eine allgemeine Formel zur schnellen Herleitung! Diese visuellen Beweise sollten in Unis und Schulen gezeigt werden! Das hilft nicht nur enorm beim Verständnis, es erregt auch das ästhetische Interesse für die Mathematik!

Did you make that Euler Maclaurin video?

I’m an amateur. I stumbled on a way to find the totals of the 12 days of Christmas pattern that involved an integral, started applying it to some power series sums, started researching power series and found this video. It’s wonderful! I want to follow this as far as you go with it. Little by little I’m figuring out why my method worked in the first place!

B0 = 0
B1 = -1/2

In video we replaced B1 with 1/2. Is it correct?

敬敬