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In-Class Exercise

Pb. 8.7 Verify the accuracy of truncating at the fifth element the following
Taylor series, in a domain that you need to specify, so the error is everywhere
less than one part in 10,000:
∞
a. ln(1+ x ) = ∑ ( 1− ) n + 1 x n
n = 1 n
∞
b. sin( )x = ∑ (−1 ) n x 2 n+1
n=0 ( n +2 1 )!
∞
c. cos( )x = ∑ (−1 ) n x 2 n

n=0 ( n2 )!

8.7.5 Reconstructing a Function from Its Fourier Components
From the results of Section 7.9, where we discussed the Fourier series, it is a
simple matter to show that any even periodic function with period 2π can be
written in the form of a cosine series, and that an odd periodic function can
be written in the form of a sine series of the fundamental frequency and its
higher harmonics.
Knowing the coefficients of its Fourier series, we would like to plot the
function over a period. The purpose of the following example is two-fold:

1. On the mechanistic side, to illustrate again the setting up of a two
indices problem in a matrix form.
2. On the mathematical contents side, examining the effects of trun-
cating a Fourier series on the resulting curve.

Example 8.8
M
∑ − ( 1 ) k
Plot yx( ) = C cos( kx), if C = 2 . Choose successively for M the val-
k
k
k=1 k + 1
ues 5, 20, and 40.
Solution: Edit and execute the following script M-file:

M= ;
p=500;
k=1:M;

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