Series Convergence

  • Static Fields 2024

    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 you 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.”

    1. Find the power series for the function \(f(z)=\frac{1}{1+z^2}\).
    2. Using the Geogebra applet from class as a model, or some other computer algebra system like Mathematica or Maple, plot both the original function and the power series to 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 your 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).”