{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# TAYLOR - EC3070 FINANCIAL DERIATIVES TAYLORS THEOREM AND...

This preview shows pages 1–2. Sign up to view the full content.

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 ) = 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}