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 £© Section
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 £© Taylor Series
In this section we will put together much of the work of Sections - in the context of
a discussion of Taylor series. We begin with two definitions.
Definition If f is a function such that f (n) is continuous on an open interval (a, b) for
n = 0, 1, 2, . . ., then we say f is C∞ on (a, b).
Definition If f is C∞ on an interval (a, b) and c is a point in (a, b), then the power
series
∞
f (n)(c) f 00(c) f 000(c)
(x − c)n = f(c) + f 0(c)(x − c) + (x − c)2 + (x − c)3 + · · · ()
X n! 2! 3!
n=0
is called the Taylor series for f about c.
A Taylor series is a power series constructed from a given function in the same manner
as a Taylor polynomial. As with any power series about c, the Taylor series for a function
f about c converges at x = c, but does not necessarily converge at any other points. If it
does converge for other values of x, it will converge absolutely on an interval (c−R, c+R),
where R is the radius of convergence. However, even if the series converges at x =6 c, it
need not converge to f(x). That is, a function may be C∞ without being analytic. (See
Problem 12 of Section for an example.) If the Taylor series does converge to f(x) for
all x in the interval of convergence, then it is the unique power series representation