Infinite Series - HP F2226A - 48GII Graphic Calculator User Manual

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Infinite series

An infinite series has the form
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
using a Taylor's series, namely,
f
(n)
where f
(x) represents the n-th derivative of f(x) with respect to x, f
If the value x
is zero, the series is referred to as a Maclaurin's series, i.e.,
0
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
order of a residual, R
(x), such that
k
n
h
(
n
( )
x
a
)
. The infinite series typically
n
0
1 ,
(
n
)
f
(
x
)
o
(
x
)
(
x
n
!
n
0
(
n
)
f
(
) 0
f
(
x
)
n
!
n
0
n
x
)
,
o
(0)
(x) = f(x).
n
x
(x), and estimate the
k
Page 13-23
by
0

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