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. 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 :) 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: https://tinyurl.com/y49buyak Enjoy :) Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer