Maclaurin
series  some mathematical experiments with Matlab

Maclaurin series are
fast approximations of functions, and
they offer more accurate function approximations than just linear ones. You
have
to consider only one general formula and you can approximate even
complicated
function values.
Maclaurin series are simpler than Taylor’s, but Maclaurin’s
are, by definition, centered
at x =
0. 
If you’re approximating a
function value for an xvalue far
from 0, you’ll have to use the slightly more
complicated Taylor series,
which work almost exactly like Maclaurin series,
except that you can center them at any xvalue.
Mathematically, we say
the Maclaurin series approximates
values of f(x)
very accurately, as long
as those xvalues are close to 0.
When you work with these
sequences, you’ll actually be
generating Maclaurin polynomials, which are some finite terms of the
series. Before
you start generating the terms to be added, make sure you understand
the
formula.
In case you don’t know
what f^{(n)}(0)
means,
it is the n^{th}
derivative
of f(x)
evaluated at 0. For example, f^{(4)}(0)
is the fourth derivative of f(x) with x
= 0.
The Maclaurin series
generates good approximations of f(x),
as long as
x
is close to 0.
You won’t use an infinite series to calculate the approximation. The
more terms
you use, however, the better your approximation will be.
As an example, let’s use
the Maclaurin polynomial
(with just
four terms in the series) for the function f(x) = sin(x) to approximate sin(0.1).
We first write the terms
of the series from n = 0 to n = 3.
And we’ll need to take
four derivatives of the function and
then evaluate them at x = 0.
Definition 1: f^{(0)}(x)
is the function itself. There’s no
derivative.
Definition 2:
0! = 1.
f^{(0)}(x)
= sin(x)
f^{(0)}(0) = sin(0) = 0
f^{(1)}(x)
= cos(x) f^{(1)}(0)
= cos(0) = 1
f^{(2)}(x)
= sin(x) f^{(2)}(0)
= sin(0) =
0
f^{(3)}(x)
= cos(x) f^{(3)}(0)
= cos(0) =
1
Let’s create a script in
Matlab to evaluate the series just
with 4 terms (0 to 3):
clear,
clc
%
Let's see more decimals
format long
%
We go from n = 0 to n = 3
n = 0 :
3;
%
This is the point for evaluation
x = 0.1;
%
These are the derivatives for each term
d = [0 1
0 1];
%
We form the sequence, following the formula
seq = d
.* x.^n ./(factorial(n))
%
We addup to get the Maclaurin approximation
approx =
sum(seq)
%
Let's compare with the official number
real_value
= sin(x)
The response is:
seq = 0
0.
0 0.67
approx =
0.33
real_value =
0.83
We’ve got almost the
exact value by using only 4 terms of
the series (and two
values are zero, because the derivative is zero for
those
terms). If we had used n = 8, it
would have been an even better approximation.
Now, let’s develop (code)
an
automated series to express the sine
function (centered at 0) using the Maclaurin expansion and let’s
compare the
results with different number of terms included.
function smp =
maclaurin_sin(x, n)
%
x = a vector with values around 0
%
n = number of terms in the series
% Start
the series with 0
smp = 0;
%
Consider all the possible derivatives
deriv =
[0 1 0 1]';
%
Iterate n times (from 0 to n1)
for i = 0 :
n1
% Implement the Maclaurin
expansion
t(i+1, :) = deriv(1) * x.^(i) /
factorial(i);
% Find the derivative for the
next term
deriv = circshift(deriv, 1);
end
%
Addup the calculated terms
smp =
sum(t);
Note:
the above code is not very efficient, because we’re calculating and
adding
terms even when their derivative is zero, which is a waste of 50% of
the time,
more or less. We’ll leave it asis just to keep and show the natural
flow of
the formula...
Now,
let’s create a script to test and use the function above.
clear,
clc, close all
%
Work with two full periods
x =
2*pi : .1 : 2*pi;
y =
sin(x);
%
Plot the goal
plot(x,
y, 'g', 'Linewidth', 3)
title('Study
of Maclaurin series')
xlabel('x')
ylabel('sin(x)
with different number of terms')
axis([2*pi
2*pi 3 3])
grid on
hold on
%
Consider 4 terms
smp =
maclaurin_sin(x, 4);
plot(x,
smp, 'ro')
%
Consider 6 terms
smp =
maclaurin_sin(x, 6);
plot(x,
smp, 'b.')
%
Consider 10 terms
smp =
maclaurin_sin(x, 10);
plot(x,
smp, 'k', 'linewidth', 3)
%
Label the calculated lines
legend('sin(x)', '4term
series', ...
'6 terms', '10
terms')
The results are:
We see that all of the
expansions work well when we are
close to 0. More terms approximate better a larger portion of the sine
curve.
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'Maclaurin
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