The Maclaurin series of a function f is the series
where f(n)(0) indicates the nth derivative of f evaluated at 0.
To expand a function f (x) in a Maclaurin series (power series about x = 0), choose
Power Series, specify desired Number
of Terms, and then specify Expand in Powers of x. With
f (x) = and 10 terms, the result is as follows.
Power Series
= 1 -
x2 +
x4 -
x6 +
x8 + O
x9
![]()
The
Ox9
term indicates that all the remaining terms in
the series contain at least x9 as a factor. (In fact, the truncation
error is of order x10 in this case.) The odd powers of x have
coefficients of 0.
Plot 2D provides an excellent visual comparison between a function and an approximating polynomial.
Plot 2D + Rectangular
, 1 -
x2 +
x4
dtbpF3in2.0003in0pt
To determine which graph corresponds to which equation, evaluate one of the
expressions where the graphs show some separation. For example,
= - .1892006238, and hence the graph of
is
the one that is negative at x = 4.
The following are additional examples of Maclaurin series expansions.
Power Series
ln(1 - x) = - x -x2 -
x3 -
x4 -
x5 + O
x6
![]()
tan-1x = x -x3 +
x5 -
x7 +
x9 + O
x10
![]()
sin2x = x2 -x4 +
x6 -
x8 + O
x10
![]()
ex = 1 + x +x2 +
x3 +
x4 +
x5 +
x6 + O
x7
![]()
sin x = x -x3 +
x5 -
x7 +
x9 + O
x10
![]()
exsin x = x + x2 +x3 -
x5 -
x6 + O
x7
![]()
Remember that output can be copied and pasted (with ordinary word-processing
tools) to create input for further calculations. In particular, select and
delete the
+ Oxn
expression to convert the series into a
polynomial. It is reassuring to note that, if the first few terms of the
Maclaurin series for ex are multiplied by the first few terms of the
Maclaurin series for sin x, then the result is the same as the first few
terms of the Maclaurin series for
exsin x.
Expand, Polynomials + Sort
1 + x +
x2 +
x3 +
x4 +
x5
x -
x3 +
x5
![]()
= x +x3 -
x5 + x2 -
x6 -
x7 +
x9 +
x10
=x10 +
x9 -
x7 -
x6 -
x5 +
x3 + x2 + x