Power Series/Euler's Great Formula

A special power series is e^x = 1 + x + x^2 / 2! + x^3 / 3! + … + every x^n / n!
The series continues forever but for any x it adds up to the number e^x

If you multiply each x^n / n! by the nth derivative of f(x) at x = 0, the series adds to f(x)
This is a TAYLOR SERIES.    Of course all those derivatives are 1 for e^x.  

Two great series are cos x = 1 - x^2 / 2! + x^4 / 4! … and sin x = x - x^3 / 3! ….
cosine has even powers, sine has odd powers, both have alternating plus/minus signs

Fermat saw magic using i^2 = -1   Then  e^ix  exactly matches   cos x + i sin x.

Professor Strang’s Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.