Power Series

Prerequisites

The students will need to know some VERY basic Mathematica or Maple. The following worksheet may be helpful.

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In-class Content

Note: if you have some students who have previously taken PH335 (Techniques of Theoretical Mechanics), they may have done the activities associated with this lecture, but it will be at least a year old. It is therefore recommended that you structure groups so that students who have previously completed the activity are grouped separately.

Homework for Symmetries

  1. (SeriesNotation1–Practice)

    Write out the first four nonzero terms in the series:

    1. $$\sum\limits_{n=0}^\infty {1\over n!}$$

    2. $$\sum\limits_{n=1}^\infty {(-1)^n\over n!}$$

  2. (SeriesNotation2–Practice)

    Write the following series using sigma $\left(\sum\right)$ notation.

    1. $$1 - 2\,\theta^2 + 4\,\theta^4 - 8\,\theta^6 +\,\dots$$

    2. $${1\over4} - {1\over9} + {1\over16} - {1\over 25}+\,\dots$$

  3. (SeriesNotation3–Practice)

    If you need more practice with sigma $\left(\sum\right)$ notation, you can get really good practice by going back and forth between the two representations of the standard power series on the memorization page. Power series are used everywhere in physics and it is very important to be able to translate back and forth between the two representations.

  4. (SeriesConvergence)

    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, 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).


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