Power Series/Euler's Great Formula
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Video Lecture 15 of 18
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### Lecture Description

In this course, Prof. Gilbert Strang discusses Power Series and 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.

### Course Description

Highlights of Calculus is a series of short videos that introduces the basics of calculus—how it works and why it is important. The intended audience is high school students, college students, or anyone who might need help understanding the subject.

The series is divided into three sections:
Introduction
- Why Professor Strang created these videos
- How to use the materials

Highlights of Calculus
- Five videos reviewing the key topics and ideas of calculus
- Applications to real-life situations and problems
- Additional summary slides and practice problems

Derivatives
- Twelve videos focused on differential calculus
- More applications to real-life situations and problems
- Additional summary slides and practice problems

Acknowledgements
Special thanks to Professor J.C. Nave for his help and advice on the development and recording of this program.The video editing was funded by the Lord Foundation of Massachusetts.