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where ξ is a number near x = x
0
. Since ξ is typically unknown, instead of an
estimate of the residual, we provide an estimate of the order of the residual in
reference to h, i.e., we say that R
k
(x) has an error of order h
n+1
, or R ≈ O(h
k+1
).
If h is a small number, say, h<<1, then h
k+1
will be typically very small, i.e.,
h
k+1
<<h
k
<< …<< h << 1. Thus, for x close to x
0
, the larger the number of
elements in the Taylor polynomial, the smaller the order of the residual.
Functions TAYLR, TAYLR0, and SERIES
Functions TAYLR, TAYLR0, and SERIES are used to generate Taylor polynomials,
as well as Taylor series with residuals. These functions are available in the
CALC/LIMITS&SERIES menu described earlier in this Chapter.
Function TAYLOR0 performs a Maclaurin series expansion, i.e., about X = 0, of
an expression in the default independent variable, VX (typically ‘X’). The
expansion uses a 4-th order relative power, i.e., the difference between the
highest and lowest power in the expansion is 4. For example,
Function TAYLR produces a Taylor series expansion of a function of any variable
x about a point x = a for the order k specified by the user. Thus, the function has
the format TAYLR(f(x-a),x,k). For example,
Function SERIES produces a Taylor polynomial using as arguments the function
f(x) to be expanded, a variable name alone (for Maclaurin’s series) or an
expression of the form ‘variable = value’ indicating the point of expansion of a
Taylor series, and the order of the series to be produced. Function SERIES
returns two output items a list with four items, and an expression for h = x - a, if
the second argument in the function call is ‘x=a’, i.e., an expression for the