Today we derive them all, the most famous infinite pi formulas: The Leibniz-Madhava formula for pi, John Wallis’s infinite product formula, Lord Brouncker’s infinite fraction formula, Euler’s Basel formula and it’s infinitely many cousins. And we do this starting with one of Euler’s crazy strokes of genius, his infinite product formula for the sine function.
This video was inspired by Paul Levrie’s one-page article Euler’s wonderful insight which appeared in the Mathematical Intelligencer in 2012. Stop the video at the right spot and zoom in and have a close look at this article or download it from here https://link.springer.com/journal/283/34/4 Very pretty.
If you are a regular and some of what I talk about in this video looks familiar that’s not surprising since we’ve visited this territory before in Euler’s real identity NOT e to the i pi = -1: https://youtu.be/yPl64xi_ZZA
1:49 A sine of madness. Euler’s ingenious derivation of the product formula for sin x
7:43 Wallis product formula for pi: pi/2 = 2*2*4*4*6*6*…/1*3*3*5*5*…
9:16 Leibniz-Madhava formula for pi: pi/4=1-1/3+1/5-1/7+…
11:50 Brouncker’s infinite fraction formula for pi: 4/pi = …
18:31 Euler’s solution to the Basel problem: pi^2/6=1/1^2+1/2^2+1/3^2+…
21:51 More Basel formulas for pi involving pi^4/90=1/1^4+1/2^4+1/3^4+… , etc.
Music (all from the free audio library that YouTube provides to creators):
Take me to the Depth (chapter transitions)
Fresh fallen snow
English country garden