Taylor series

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This article/section deals with mathematical concepts appropriate for late high school or early college.

The Taylor series of a function is useful for approximating a mathematical function near to some particular point. For a function , the Taylor series about the point is

where each of the derivatives is to be evaluated at . If as the series converges, then it is exact. Otherwise, it can be used as an approximation. Often, Taylor series are performed around , in which case they are sometimes also known as a Maclaurin series.

Examples of common Taylor series

Extensions of the Exponential Function

Consider the exponential of imaginary number ,

By the power laws then all complex numbers have an exponential,