TaylorsTheoremTimeless

# TaylorsTheoremTimeless - Econ 204 Taylor's Theorem In this...

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Econ 204 Taylor’s Theorem In this supplement, we give alternative versions of Taylor’s Theorem. For univariate functions, we provide a different formulation of the error term using so-called “little oh” and “big Oh” notation. For multivariate functions, we provide the quadratic form of Taylor’s Theorem (de la Fuente just provides the linear form, with quadratic error term) and analyze it as a quadratic form using the machinery in the Supplement to Section 3.6. Definition 1 We say y = o ( x ) as x 0 if | y | | x | 0 as x 0 and y = O ( x ) as x 0 if | y | | x | is bounded as x 0 or more formally M ε> 0 | x | ≤ ε ⇒ | y | ≤ M | x | The following theorem is a consequence of Theorem 1.9 on page 160 of de la Fuente. In my experience, knowing the exact form of the error term E n as given in de la Fuente is not particularly useful, because one does not know in advance the location of x + λh at which E n is evaluated. However, if f has an ( n + 1) st derivative which is continuous, one can obtain a O ( h n +1 ) error term from the formula for E n . Theorem 2 (Taylor’s Theorem for Univariate Functions) Let f : I R be n -times differentiable, where I R is an open interval. If x I , then f ( x + h ) = f ( x ) + n k =1 f ( k ) ( x ) h k k !

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