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Taylor and Maclaurin Series

3
4
2
)
)
)
kV cos
ωt
kV cos
kV cos
ωt
ωt
⎛
p
p
p
i = a kV cos ωt + ( -------------------------------- + ( -------------------------------- + ( -------------------------------- + … ⎞ (6.117)
⎝ max 2! 3! 4! ⎠
This expression can be simplified with the use of the following trigonometric identities:
2 1 1
-
-
cos x = -- + -- cos 2x
2 2
-- cos
cos 3 x = 3 - x 1 - 3x (6.118)
-- cos +
4 4
cos 4 x = 3 - 1 - 2x + 1 - 4x
-- +
-- cos
-- cos
8 2 8
Then, substitution of (6.118) into (6.117) and after simplification, we obtain a series of the fol-
lowing form:
i = A + A cos ωt + A cos 2ωt + A cos 3ωt + A cos 4ωt + … (6.119)
3
1
0
4
2
We recall that the series of (6.119) is the trigonometric series form of the Fourier series. We
observe that it consists of a constant term, a term of the fundamental frequency, and terms of all
harmonic frequencies, that is, higher frequencies which are multiples of the fundamental fre-
quency.
6.10 Taylor and Maclaurin Series
A function fx() which possesses all derivatives up to order at a point x = x 0 can be expanded
n
in a Taylor series as
f'' x ( ) f n () x ( )
fx() = fx ) ( 0 f' x ) ( + 0 ( xx ) – 0 -------------- x –( + 2! 0 x ) 0 2 + … + ------------------- x –( n! 0 x ) 0 n (6.120)
If x = 0 , (6.120) reduces to
0

n ()
f 0 ()
fx() = f0() + f' 0()x + f'' 0() 2 … + -----------------x n (6.121)
------------x +
2! n!
Relation (6.121) is known as Maclaurin series, and has the form of power series of (6.110) with

⁄
a = f n () 0 () n! .
n
To appreciate the usefulness and application of the Taylor series, we will consider the plot of Fig-
(
ure 6.32, where iv() represents some experimental data for the current−voltage i – v ) characteris-
tics of a semiconductor diode operating at the 0 ≤≤ 5 volts region.
v

Numerical Analysis Using MATLAB® and Excel®, Third Edition 6−41
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