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Page 13-23
Taylor and Maclaurin’s series
A function f(x) can be expanded into an infinite series around a point x=x
0
by
using a Taylor’s series, namely,
,
where f
(n)
(x) represents the n-th derivative of f(x) with respect to x, f
(0)
(x) = f(x).
If the value x
0
is zero, the series is referred to as a Maclaurin’s series, i.e.,
Taylor polynomial and reminder
In practice, we cannot evaluate all terms in an infinite series, instead, we
approximate the series by a polynomial of order k, P
k
(x), and estimate the order
of a residual, R
k
(x), such that
,
i.e.,
The polynomial P
k
(x) is referred to as Taylor’s polynomial. The order of the
residual is estimated in terms of a small quantity h = x-x
0
, i.e., evaluating the
polynomial at a value of x very close to x
0
. The residual if given by
,
∑
∞
=
−⋅=
0
)(
)(
!
)(
)(
n
n
o
o
n
xx
n
xf
xf
∑
∞
=
⋅=
0
)(
!
)0(
)(
n
n
n
x
n
f
xf
∑∑
∞
+==
−⋅+−⋅=
1
)(
0
)(
)(
!
)(
)(
!
)(
)(
kn
n
o
o
n
k
n
n
o
o
n
xx
n
xf
xx
n
xf
xf
).()()( xRxPxf
kk
+=
1
)1(
!
)(
)(
+
+
⋅=
k
k
k
h
k
f
xR
ξ