ECON211 Chapter 11: Taylor Series
27 Jul 2016
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The taylor series for f (x) about point a is f (x) = f (a) + (x a) f (a) (x a)2 f (a) 1! (x a)n f (n) (z) n! Notice that f (n) in rn is evaluated at z, and not at a. Now suppose we want to solve some nonlinear function f for the value of x, say x , such that f (x ) = 0. Let a be the initial guess at the solution. Then from above a taylor series with n = 2 yields f (x) = f (a) + (x a) f (a) Setting f (x) = r2 = 0 and solving for x we obtain x = a f (a) f (a) x is the next step in the process of searching for the solution x . Under quite general conditions this algorithm converges to the solution. The idea can be generalized to n nonlinear functions in n variables.
