assignment Homework
Power Series Sequence (E&M)
Static Fields 2023 (6 years)
Recall that, if you take an infinite number of terms, the power 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 what it means for the power
series to “converge” for all \(z\). Not all power series converge for all values
of the argument of the function. More
commonly, a power series is only a valid, equivalent representation of a
function for some more restricted values of \(z\), EVEN IF YOUR KEEP AN INFINITE NUMBER OF TERMS. The technical name
for this idea is convergence--the series only "converges" to the
value of the function on some restricted domain, called the “interval” or “region of convergence.”
Find the power series for the function \(f(z)=\frac{1}{1+z^2}\). Then,
using the Geogebra applet from class as a model,
or some other computer algebra system like Mathematica 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. You may need to include a lot of terms to see the effect of the region of convergence. You may also need to play with the values of \(z\) that you plot. Keep adding terms until you see a really strong effect!
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).”