HP 50g Graphing Calculator - Manual (Page 458)

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
x x
n
x f
x f
∑
∞
=
⋅ =
0
) (
!
) 0 (
) (
n
n
n
x
n
f
x f
∑ ∑
∞
+ = =
− ⋅ + − ⋅ =
1
) (
0
) (
) (
!
) (
) (
!
) (
) (
k n
n
o
o
n k
n
n
o
o
n
x x
n
x f
x x
n
x f
x f
). ( ) ( ) ( x R x P x f
k k
+ =
1
) 1 (
!
) (
) ( +
+
⋅ = k
k
k
h
k
f
x R
ξ