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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

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>

<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.

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.