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Taylor Series convergence intervals w/o remainder
Posted:
May 11, 1998 11:42 AM


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://wwwcm.math.uiuc.edu http://netmath.math.uiuc.edu



