Day 49 in MAT229

Today:

• Announcements:
  • Sorry, I haven't graded your 11.6 homework yet.
  • Your 11.7 homework is due today.

• Section 11.9: Representations of Functions as Power Series
  • Summary:

    Once we have one power series, we have an infinite number of other power series. We can use function composition to create others. [If you didn't have any respect for composition before, I hope you have some now!:)]

    So, for example, from , we can produce series for

    1. 
    2. 
    3. 

    The other very important notion discussed here is the validity of differentiation and integration of power series. And the big news is that both operations are legitimate on the interval of convergence of the original power series:

    

    Hence, we can also generate the following power series from the series for :

    1. What do you get when you integrate ? How can we find a power series for this function?
    2. Given the power series for : what would be the power series of its anti-derivative(s)?

  • Examples:
    • What do we get when we differentiate the series for term-by-term?
    • #5, p. 775
    • #9
    • #19
    • #23
    • #29
    • #41

• Section 11.10: Taylor and MacLaurin Polynomials
  • The big picture: we're approximating functions (some nasty) with nice ones (polynomials):

    

    

    Let's talk error now. Remember error? As we discussed for the case of Simpson's rule, it's nice to give an estimate -- but even nicer when you can provide an error estimate, also....

    
    
    

  • Examples:
    • Let's find the Taylor series for ln (which is one that Nick asked about) around 3. That is, we're looking for values of for arguments of around . We can do it in two different ways:
      1. using our favorite geometric series, and ; or
      2. using Taylor polynomials, and .

    • Taylor Polynomials Lab (from Stewart's Calculus)

• Worksheets from the University of Arizona