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Unformatted text preview: EC3070 FINANCIAL DERIATIVES TAYLOR’S THEOREM AND SERIES EXPANSIONS Taylor’s Theorem. If f is a function continuous and n times differentiable in an interval [ x, x + h ], then there exists some point in this interval, denoted by x + λh for some λ ∈ [0 , 1], such that f ( x + h ) = f ( x ) + hf ( x ) + h 2 2 f ( x ) + · · · · · · + h ( n − 1) ( n − 1)! f ( n − 1) ( x ) + h n n ! f n ( x + λh ) . If f is a so-called analytic function of which the derivatives of all orders exist, then one may consider increasing the value of n indefinitely. Thus, if the condition holds that lim n →∞ h n n ! f n ( x ) = 0 , which is to say that the terms of the series converge to zero as their order increases, then an infinite-order Taylor-series expansion is available in the form of f ( x + h ) = ∞ X j =0 h j j ! f j ( x ) . This is obtained simply by extending indefinitely the expression from Taylor’s Theorem. In interpreting the summary notation for the expansion, one must be aware of the convention that 0! = 1. A Taylor-series expansion is available for functions which are analytic within a restricted domain. An example of such a function is (1 − x ) − 1 . The function and its derivatives are undefined at the point x = 1. Nevertheless, Taylor-series expansions exists for the function at all other points and for all | h | < 1. Another example is provided by the function log( x ) which is defined...
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This note was uploaded on 03/02/2012 for the course EC 3070 taught by Professor D.s.g.pollock during the Spring '12 term at Queen Mary, University of London.
- Spring '12