Power Series/Euler's Great Formula | MIT Highlights of Calculus

Power Series/Euler's Great Formula
Instructor: Gilbert Strang
http://ocw.mit.edu/highlights-of-calculus

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.

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