Taylor series

The Taylor series of a function is useful for approximating a mathematical function near to some particular point. For a function <math>f(x)</math>, the Taylor series about the point <math>x_0</math> is <math>f(x-x_0)=f(x_0)+x_0\frac{df}{dx}+\frac{x_0^2}{2!}\frac{d^2f}{dx^2}+\ldots+\frac{x_0^N}{N!}\frac{d^Nf}{dx^N}</math> where each of the derivatives is to be evaluated at <math>x=x_0</math>. If <math>N\rightarrow\infty</math> the series is exact, otherwise it is an approximation. Often, Taylor series are performed around <math>x_0=0</math>, in which case they are sometimes also known as a Maclaurin series.

Examples of common Taylor series

<math>e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots</math>

<math>\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\ldots</math>

<math>\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots</math>