Day 36 in MAT229

Last time

Next time

Today:

Announcements:
- Homework 11.4 graded.
  - 20
  - 30
  - 40
- Homework 11.5 is due.
- Tomorrow's quiz opportunity will again be over 11.8: power series
- You have an exam on Thursday. It will be through 11.7, which is the summary chapter that we'll consider today.
Section 11.7: Strategies for testing series
- Examples: let's take a look at the odd numbers, p. 764. Which do you know how to do? Which not so much?
Section 11.8: Power series
- Summary:
  Now, rather than simply infinite sums of numbers, we're going to consider an infinite sum of functions in what are known as power series. They're like polynomials, but they don't have finite degree.
  The terms will be monomials -- constants times powers of x.
  Now we realize that, with functions for terms, each value of x specified gives rise to an infinite series, and we might immediately wonder if the series is convergent.
  So what kind of function has an infinite number of terms? How do you evaluate such a thing? What kinds of function are these? It turns out that a lot of our old friends can be expressed this way
  ${F(x)=e^x}$ , for example, as we've seen.
  Now what did we assert about ${F(x)=e^x}$ ? That
  ${e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}}$
  For what values of x would this converge?
  Apply the ratio test for arbitrary x, and what do you discover?
  
  So ${F(x)=e^x}$ presents us with a case where we have an "infinite radius of convergence".
  Let's look at an example where our radius of convergence is not so nice: consider ${F(x)=\frac{1}{1-x}}$ . One can show that
  
  ${\frac{1}{1-x}=\sum_{n=0}^{\infty}\qquad{x^n}}$ In fact we deduced this for a particular case when we looked at geometric series: that ${\frac{1}{1-r}=\sum_{n=0}^{\infty}\qquad{r^n}}$ But now we're talking about a function of x. One can think of x as representing allowable values of r (that is, values of r for which this geometric series converges). We know that this is for values of r such that ${|r|<1}$ .
  
  Now: for what values of x do you think that this will converge?
  
  Note that this is a variant on the theme of the ratio test: we don't consider the power terms, but only the coefficients.
- Examples:
  - What does Theorem 2 say will be the radius of convergence of the series for ${F(x)=\frac{1}{1-x}}$ ?
  - What about the endpoints?
  - #7, p. 769
  - #19
  - #27

Website maintained by Andy Long. Comments appreciated.