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Topic: Taylor Series convergence intervals w/o remainder
Replies: 1   Last Post: May 14, 1998 11:49 AM

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Jerry Uhl

Posts: 1,267
Registered: 12/3/04
Taylor Series convergence intervals w/o remainder
Posted: May 11, 1998 11:42 AM
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John Graham makes a very good point: it is not enough to know the Taylor
series convergence, a serious erro committed by many calculus courses and
apparently by the AP board.

There is a simple way to get the point across without terrible calculations.

In Calculus&Mathematica, we have had a good deal of success in approaching
Taylor Series convergence intervals via complex numbers - the way it is
done in advanced math courses.

For instance the convergence interval for the expansion of
1/(1 + x^2) in powers of (x - 0) is (-1,1) because of complex
singularities at x = Sqrt[-1] and x = -Sqrt[-1] which are distance 1 from 0
in the complex plane.

the convergence interval for the expansion of
1/(1 + x^2) in powers of (x - b) is (b - Sqrt[b^2 + 1],2 + Sqrt[b^2 + 1])
because of complex singularities at x = Sqrt[-1] and x = -Sqrt[-1] which
are distance Sqrt[b^2 + 1] from b in the complex plane.

Nasty functions such as e^(-1/x^2) are no problem here because 0 itself is
a complex (not real) singularity of e^(-1/x^2). (put x = Sqrt[-1] t, and
let t->0). This is the sinal that Taylor Seres does not work for this
function.

When you give your students this criterion, you are giving them something
they can use. And it's a signal that there is interesting math for them in
the future.

And it eliminates the need for the remainder term.

-Jerry Uhl




----------------------------------------------------------------------
Jerry Uhl juhl@ncsa.uiuc.edu
Professor of Mathematics 1409 West Green Street
University of Illinois Urbana,Illinois 61801
Calculus&Mathematica Development Team

Any fool can know. The point is to understand.
------Albert Einstein

http://www-cm.math.uiuc.edu
http://netmath.math.uiuc.edu










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