Maclaurin Series

A Maclaurin series is a Taylor series expansion of a function about 0,

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Maclaurin series are named after the Scottish mathematician Colin Maclaurin.

The Maclaurin series of a function up to order may be found using Series[f, x, 0, n]. The th term of a Maclaurin series of a function can be computed in the Wolfram Language using SeriesCoefficient[f, x, 0, n] and is given by the inverse Z-transform

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Maclaurin series are a type of series expansion in which all terms are nonnegative integer powers of the variable. Other more general types of series include the Laurent series and the Puiseux series.

Maclaurin series for common functions include

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The explicit forms for some of these are

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where is a gamma function, is a Bernoulli number, is an Euler number and is a Legendre polynomial.