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Infinite series
An infinite series has the form
n
n
axnh )()(
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
xx
n
xf
xf
,
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
xf
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
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