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Taylor Series - Definition
9.11 TAYLOR SERIES DEFINITION If we continue the Taylor polynomial (by adding three dots at the end) we obtain a power series f(c) + f'(c)(x - c) +
This series is called the Taylor series for the function f(x) about the point x = c. The Taylor series about the point x = 0 is called the MacLaurin series, f(0) + f'(0)x + At x = c the Taylor series about the point c converges to f(c). But we have no assurance that the Taylor series converges to f(x) at any other point x. There are three possibilities and all of them arise: (1) The Taylor series diverges at x. (2) The Taylor series converges but to a value different than f(x). (For an example, see Problem 28 at the end of this section.) (3) The Taylor series converges to f(x); i.e., f(x) is equal to the' sum of its Taylor series.
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(x -c)2 + ... +
(x - c)n + ... 
+ ...+ 
+....
Last Update: 2006-11-08