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{{Template:Math-h}}

{{Math-h}}

The '''Taylor series''' of a function is useful for approximating a [[function|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 [[function|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

where each of the derivatives is to be evaluated at <math>x=x_0</math>. If as <math>N\rightarrow\infty</math> the [[series (mathematics)|series]] converges, then it is exact. Otherwise, it can be used as 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 as <math>N\rightarrow\infty</math> the [[series (mathematics)|series]] converges, then it is exact. Otherwise, it can be used as 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===

==Examples of common Taylor series==

<math>e^x=\sum_{n=0}^{\infin} \frac{x^{n}}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots</math>

<math>e^x=\sum_{n=0}^{\infin} \frac{x^{n}}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots</math>

<math>\cos x=\sum_{n=0}^{\infin} (-1)^n\frac{x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots</math>

<math>\cos x=\sum_{n=0}^{\infin} (-1)^n\frac{x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots</math>

====Extensions of the Exponential Function====

===Extensions of the Exponential Function===

Consider the exponential of [[imaginary number]] <math>yi</math>,

Consider the exponential of [[imaginary number]] <math>yi</math>,

<math>e^{x+yi}=e^{x}(\cos y+i\sin y)</math>.

<math>e^{x+yi}=e^{x}(\cos y+i\sin y)</math>.

[[category:calculus]]

[[Category:Calculus]]