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/