Data collected by Prof. Enrique Mateus NievesPhD in Mathematics Education.

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

(1)

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 Mathematica

using SeriesCoefficient[f, x, 0, n ] and is given by the inverse Z-transform

(2)

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(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

Data collected by Prof. Enrique Mateus NievesPhD in Mathematics Education.

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

The explicit forms for some of these are(34)

(35)

(36)

(37)

(38)

Data collected by Prof. Enrique Mateus NievesPhD in Mathematics Education.

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

where is a gamma function, is a Bernoulli number, is an Euler number and

is a Legendre polynomial.

Data collected by Prof. Enrique Mateus NievesPhD in Mathematics Education.

SEE ALSO: Alcuin's Sequence, Fourier Series, Generalized Fourier Series, Lagrange

Inversion Theorem, Lagrange Remainder, Laurent Series, Power Series, Puiseux

Series, Series Expansion, Taylor Series

REFERENCES:

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton,

FL: CRC Press, pp. 299-300, 1987.

Referenced on Wolfram|Alpha: Maclaurin Series.

CITE THIS AS:

Weisstein, Eric W. "Maclaurin Series." From MathWorld--A Wolfram Web Resource.

http://mathworld.wolfram.com/MaclaurinSeries.html