Euler MacLaurin Summation Formula

Derivation of the Euler MacLaurin Summation Formula following Euler's methods, Introduction to the Bernoulli numbers, Application of the EMSF to Faulhaber's Formula and the Basel Probem

[@abbasfadhil1715]

First, and phenomenal vid i especially like the derivation vids 🫡

[@FireSermon83]

Thanks so much for the positive feedback! I’m glad you enjoyed the video.

[@eiseks3410]

The coefficients in front of the derivatives of f(x) in the formula of S'(x) are clearly values of the Zeta function at negative integers (and at zero). Has this been proven rigorously?

[@FireSermon83]

@@eiseks3410 Thanks for your question. You are absolutely correct in observing that the Bernoulli numbers are very closely related to values of the zeta function. I have a video coming up about this. Stay tuned!

[@nin10dorox]

This is great! Do you have a source linking this derivation to Euler? I've been having trouble finding any papers laying out his method.

[@FireSermon83]

@@nin10dorox I’m glad you enjoyed the video. Here are a couple sources that you might find helpful if you want to see Euler’s original argument: arxiv.org/pdf/0806.4096 And: www.ms.uky.edu/~sohum/ma330/files/euler_zeta_ayoub.pdf