TOPICS
Maclaurin Series
A Maclaurin series is aTaylor series expansionof 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 usingSeries[f,
x, 0,n
]. The
th term of a Maclaurin series of a function
can be computed in theWolfram Language usingSeriesCoefficient[f,
x, 0,n
] and is given by the inverseZ-transform
.](/image.pl?url=http%3a%2f%2fmathworld.wolfram.com%2fimages%2fequations%2fMaclaurinSeries%2fNumberedEquation2.svg&f=jpg&w=240) | (2) |
Maclaurin series are a type ofseries expansion in which all terms are nonnegative integer powers of the variable. Other more general types of series include theLaurent series and thePuiseux series.
Maclaurin series for common functions include
 | (3) |
 | (4) |
 | (5) |
 | (6) |
 | (7) |
 | (8) |
 | (9) |
 | (10) |
 | (11) |
 | (12) |
 | (13) |
 | (14) |
 | (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) |
 | (39) |
 | (40) |
 | (41) |
 | (42) |
 | (43) |
 | (44) |
 | (45) |
 | (46) |
 | (47) |
 | (48) |
 | (49) |
 | (50) |
 | (51) |
 | (52) |
 | (53) |
where
is agamma function,
is aBernoulli number,
is anEuler number and
is aLegendre polynomial.
See also
Alcuin's Sequence,Fourier Series,Generalized Fourier Series,Lagrange Inversion Theorem,Lagrange Remainder,Laurent Series,Power Series,Puiseux Series,Series Expansion,Taylor SeriesExplore this topic in the MathWorld classroomExplore with Wolfram|Alpha
More things to try:
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 SeriesCite this as:
Weisstein, Eric W. "Maclaurin Series."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/MaclaurinSeries.html