Power Series Expansions

$$e^{x} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots$$ (48)

$$\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \ldots$$ (49)

$$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} + \ldots$$ (50)

Depending on where you start, these can be used to prove the relations above. They are most useful for getting expansions for small values of their parameters. For small x (to leading order):

$$e^{x} \approx 1 + x$$ (51)

$$\cos(x) \approx 1 - \frac{x^2}{2!}$$ (52)

$$\sin(x) \approx x$$ (53)

$$\tan(x) \approx x$$ (54)

We will use these fairly often in this course, so learn them.