assignment Homework
Recall that, if you take an infinite number of terms, the series for
\(\sin z\) and the function itself \(f(z)=\sin z\) are equivalent
representations of the same thing for all real numbers \(z\), (in
fact, for all complex numbers \(z\)). This is not always true. More
commonly, a series is only a valid, equivalent representation of a
function for some more restricted values of \(z\). The technical name
for this idea is convergence--the series only "converges" to the
value of the function on some restricted domain.
Find the power series for the function \(f(z)=\frac{1}{1+z^2}\). Then,
using the Mathematica worksheet from class (vfpowerapprox.nb) as a model,
or some other computer algebra system like Sage or Maple, explore
the convergence of this series. Where does your series for this new
function converge? Can you tell anything about the region of
convergence from the graphs of the various approximations? Print out a
plot and write a brief description (a sentence or two) of the region
of convergence.
Note: As a matter of professional ettiquette (or in some cases, as a legal
copyright requirement), if you use or modify a computer program written by
someone else, you should always acknowledge that fact briefly in whatever you
write up. Say something like: “This calculation was based on a (name
of software package) program titled (title) originally written by
(author) copyright (copyright date).