Power Series/Euler's Great Formula

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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.

Related Resources

Lecture summary and Practice problems (PDF)

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