Taylor series

The Taylor series of a function is useful for approximating a mathematical function near to some particular point. For a function $f(x)$, the Taylor series about the point $x_{0}$ is

$f(x-x_{0})=f(x_{0})+x_{0}{\frac {df}{dx}}+{\frac {x_{0}^{2}}{2!}}{\frac {d^{2}f}{dx^{2}}}+\ldots +{\frac {x_{0}^{N}}{N!}}{\frac {d^{N}f}{dx^{N}}}$

where each of the derivatives is to be evaluated at $x=x_{0}$. If $N\rightarrow \infty$ the series is exact, otherwise it is an approximation. Often, Taylor series are performed around $x_{0}=0$, in which case they are sometimes also known as a Maclaurin series.

Examples of common Taylor series

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

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

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