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

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From 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 đ

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 đ Anyway, lots of things to look forward to: a ton of power sum formulas, animations of a couple of my favourite âproofs without wordsâ, 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.

Here is a link to a couple of slides that show how to justify having all summations in sight run from 1 to n. This is the challenge that follows the discussion at https://youtu.be/fw1kRz83Fj0?t=2440 . Had this in the video originally and then decided to make this into a challenge: http://www.qedcat.com/misc/change_limits.pdf

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 đ

Finally, check out the article "Gaussâ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: https://tinyurl.com/y49buyak

Enjoy đ

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