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The Taylor series of <math> e^x </math> is | The Taylor series of <math> e^x </math> is | ||
| − | <math> e^x = \sum^{\infty}_{n=0}{\frac{x^n}{n!}} = 1 + x + \frac{x^2}2 + \frac{x^3}6 + \cdots </math> | + | |
| + | <math> e^x = \sum^{\infty}_{n=0}{\frac{x^n}{n!}} = 1 + x + \frac{x^2}2 + \frac{x^3}6 + \cdots </math> | ||
| − | Using this equation, it is possible to relate <math>e</math> to the seemingly unrelated worlds of trigonometry and the complex numbers by simply plugging in a complex number, <math> | + | Using this equation, it is possible to relate <math>e</math> to the seemingly unrelated worlds of trigonometry and the complex numbers by simply plugging in a complex number, <math>ix</math> for example. This yields: |
| − | + | | |
| + | <math> e^ix = \sum^{\infty}_{n=0}{\frac{(ix)^n}{n!}} = \sum^{\infty}_{n=0}{\frac{i^nx^n}{n!}} = 1 + x + \frac{x^2}2 + \frac{x^3}6 + \cdots </math> | ||
But by rearranging this, one gets the identity | But by rearranging this, one gets the identity | ||
Revision as of 12:38, 2 December 2018
$ e $ and Trigonometry
The Taylor series of $ e^x $ is
$ e^x = \sum^{\infty}_{n=0}{\frac{x^n}{n!}} = 1 + x + \frac{x^2}2 + \frac{x^3}6 + \cdots $
Using this equation, it is possible to relate $ e $ to the seemingly unrelated worlds of trigonometry and the complex numbers by simply plugging in a complex number, $ ix $ for example. This yields:
$ e^ix = \sum^{\infty}_{n=0}{\frac{(ix)^n}{n!}} = \sum^{\infty}_{n=0}{\frac{i^nx^n}{n!}} = 1 + x + \frac{x^2}2 + \frac{x^3}6 + \cdots $
But by rearranging this, one gets the identity
File:Ezcis.jpg
350px
References:
(Reference 1)
(Reference 2)
