Unformatted text preview: and the true value is at most the size of the next term. A similar result is true of many Taylor series. THEOREM 10.52 Suppose that f is deﬁned on some open interval I around a and suppose f ( N +1) ( x ) exists on this interval. Then for each x ± = a in I there is a value z between x and a so that f ( x ) = N X n =0 f ( n ) ( a ) n ! ( xa ) n + f ( N +1) ( z ) ( N + 1)! ( xa ) N +1 . Proof. The proof requires some cleverness to set up, but then the details are quite elementary. We want to deﬁne a function F ( t ). Start with the equation F ( t ) = N X n =0 f ( n ) ( t ) n ! ( xt ) n + B ( xt ) N +1 ....
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 Fall '07
 JonathanRogawski
 Math, Calculus, Maclaurin Series, Taylor Series, Mathematical Series, Taylor’s Theorem

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