Taylor and Maclaurin Series
Last Revision: October 5th, 2010
Maple Commands
Formal Power Series and the coeftayl command
Maple lets you find a formal Taylor series expansion in inert form by converting the function
into a Taylor series.
Example: Expand the function
at the point c=1. First we define the function, and
then we use the convert command
f := x -> 1/x;
(1.1.1)
convert( f(x) , FormalPowerSeries, x=1, n);
(1.1.2)
The option x=1 communicates the center of the expansion, and the option 'n' selects the
counting index in the sum. Evaluation of the sum returns the original function.
value(%);
1
(1.1.3)
With the command coeftayl you can calculate a particular coefficient in the Taylor series. In the
example with
and
the coefficient associated with
is -1, and here is the
confirmation
coeftayl( 1/x,x=1,13);
(1.1.4)
coeftayl computes the quantity
and we can do that directly using the appropriate
differentiation command. In our example it is (the definition
is still in effect)
(D@@13)(f)(1)/13!;
(1.1.5)
Example: What is the coefficient associated with
in the expansion of
centered at c=0?
coeftayl( exp(-x)*sin(2*x),x=0,7);
139
2520
or in explicit computation
f := x -> exp(-x)*sin(2*x);
(1.1.6)
(1.1.7)
(D@@7)(f)(0)/7!;
139
2520
(1.1.7)
The taylor and the series commands
You can display the first few term in a series expansion with the taylor command.
Example: We use
at the point c=1 again.
taylor( 1/x,x=1);
(1.2.1)
The last expression indicates the order of the approximation. It can be controlled with a third
optional entry.
taylor(1/x, x=1, 12);
(1.2.2)
Example: We expand sin(x) at c=0. Notice that the third option does not control the number of
terms.
taylor(sin(x),x=0,8);
(1.2.3)
taylor(sin(x),x=0,9);
(1.2.4)
taylor(sin(x),x=0,10);
(1.2.5)
The series command accomplishes the same task, and it can even find series expansions where
the taylor command fails.
Example: We take again
at the point c=1 .
series( 1/x,x=1);
(1.2.6)
Example: The function
is undefined at x=0, and a Taylor series
centered at c=0 does not exist. However, the series command finds an expansion anyway.
taylor( 1/sin(x), x=0); # error, as expected
Error, does not have a taylor expansion, try series()
series( 1/sin(x),x=0);
(1.2.7)
(1.2.7)
This series expansion approximates the pace at which the function approaches
zero, as the graph below indicates.
f := x -> 1/sin(x);
as x goes to
(1.2.8)
p := x -> 1/x +x/6+7/360*x^3;
(1.2.9)
plot( [f,p],-4..4,view=[-4..4,-4..4],discont=true,color=
[red,blue]);
4
3
2
1
1
2
3
4
Direct Construction of Taylor Polynomials
We now turn to the direct construction of Taylor polynomials without the aid of explicit maple
commands. For instance, the coeftayl command just calculates the quantity
and we
can do that directly using the appropriate differentiation command. In order to construct the Taylor
polynomial of a given degree we need to
1. define the function
2. select the center c
3. select the desired degree N of the polynomial
4. assemble the polynomial
Example: Construct the Maclaurin polynomial of degree 8 for the function
now go through the steps described above.
f := x -> exp(-x/2)*sin(x);
We
(2.1)
c := 0;
N := 8;
(2.2)
(2.2)
p := x -> sum( (D@@n)(f)(c)*(x-c)^n/n!, n=0..N);
(2.3)
p(x);
#it's there
(2.4)
Here are the function and its approximation in a common graph.
plot( [f(x),p(x)], x=-5..5,color=[red,blue], view=[-5..5,-4.
.6],numpoints=2000);
6
4
2
2
4
x
Example: Find the Taylor polynomial of degree 5 centered at c=4 for the function
We modify the steps in the last example, and conclude with a graph.
.
f := x -> sqrt(x);
c := 4; N := 5;
p := x -> sum( (D@@n)(f)(c)*(x-c)^n/n!, n=0..N);
(2.5)
p(x);
(2.6)
plot([f(x),p(x)], x=-2..12, color=[red,blue],view=[-2..12,-2.
.5]);
5
4
3
2
1
2
4
6
x
8
10
12