HP 49g+ - Manual (Page 454)

Page 13-23
Infinite series
An infinite series has the form
n
n
a x n h ) ( ) (
1 , 0
−
∑
∞
=
. The infinite series typically
starts with indices n = 0 or n = 1. Each term in the series has a coefficient
h(n) that depends on the index n.
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,
∑
∞
=
− ⋅ =
0
) (
) (
!
) (
) (
n
n
o
o
n
x x
n
x f
x f
,
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.,
∑
∞
=
⋅ =
0
) (
!
) 0 (
) (
n
n
n
x
n
f
x f
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