This document provides an overview of Taylor series, including:
- Taylor series represent a way to approximate functions as infinite sums of terms calculated from the function's derivatives.
- They allow expressing any function that satisfies certain conditions as an infinite series centered around a specific point.
- The general form of a Taylor series for a function f around a point a is given as the sum of terms involving derivatives of f evaluated at a.
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2. 17/3/2016 Taylor Series from Wolfram MathWorld
http://mathworld.wolfram.com/TaylorSeries.html 2/3
(15)
Rewriting the repeated integral then gives
(16)
Now, from the meanvalue theorem for a function , it must be true that
(17)
for some . Therefore, integrating times gives the result
(18)
(Abramowitz and Stegun 1972, p. 880), so the maximum error after terms of the Taylor series is the maximum value
of (18) running through all . Note that the Lagrange remainder is also sometimes taken to refer to the
remainder when terms up to the st power are taken in the Taylor series (Whittaker and Watson 1990, pp. 95
96).
Taylor series can also be defined for functions of a complex variable. By the Cauchy integral formula,
(19)
(20)
(21)
In the interior of ,
(22)
so, using
(23)
it follows that
(24)
(25)
Using the Cauchy integral formula for derivatives,
(26)
An alternative form of the onedimensional Taylor series may be obtained by letting
(27)
so that
(28)
Substitute this result into (◇) to give
(29)
A Taylor series of a real function in two variables is given by
(30)
This can be further generalized for a real function in variables,
(31)
Rewriting,
(32)
For example, taking in (31) gives
(33)