Taylor approximation of tan^2(x)Date: 6/12/96 at 9:22:41 From: Anonymous Subject: Taylor approximation of tan^2(x) I just want to check that I can't do the Taylor approximation of tan^2(x) because f'(0)=infinity. Any guesses on how I could get an approximation (non-piecewise)? Thanks. Date: 6/13/96 at 1:54:48 From: Doctor Pete Subject: Re: Taylor approximation of tan^2(x) Perhaps it might be useful to think of Taylor series in their general form; recall that the Taylor series is taken about some point x = a, that is, oo \---\ 1 d^k | f(x) = > --- ---- f | (x-a)^k . /---/ k! dx^k |x=a k=0 Depending on the behavior of the derivatives of f, this approximation may not converge at a=0, which is the problem you are presented with. (Incidentally, the case a=0 in the Taylor series above is usually called the Maclaurin series of f(x).) So the obvious thing to do is to consider some other value for a for which the derivatives are finite. If, however, you wish to approximate numerical values for tan^2(x) near 0, you may find the convergence of the Taylor series about some nonzero point a to be rather poor, if it converges at all. -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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