Day 40 in MAT229

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Today:

Announcements:
- Your exam is graded. It didn't go that well. Two of you got As, at any rate.
  - The exam
  - The "key"
  - The curve (calculate your percentage out of 75, rather than 85).
  - Some problems we should look at:
    - Your vocabulary is too sloppy at times: many use "it" without any indication of the reference. What is "it"? Use words like "terms", "series", "integrand", etc.
- You have a homework due tomorrow.
- I pushed the homework for 11.8 back to Thursday.
- Quiz opportunity tomorrow is over 11.9.
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?
  
  Let's take a look at how well a truncated power series -- a partial sum -- approximates the real function.
  1. Graph ${e^x}$ on the interval [-2,2].
  2. Graph ${p_1(x)=\sum_{n=0}^{1}\frac{x^n}{n!}}$
  3. Graph ${p_2(x)=\sum_{n=0}^{2}\frac{x^n}{n!}}$
  4. Graph ${p_3(x)=\sum_{n=0}^{3}\frac{x^n}{n!}}$
  What do you see happening?
  
  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".
  
  Interesting sidelight: you may have encountered Euler's Formula at some point in your life:
  
  ${e^{i\theta}=\cos(\theta)+i\sin(\theta)}$
  where ${i=\sqrt{-1}}$ . From this (and the power series for ${e^x}$ ) we can now deduce power series for the functions cosine and sine. Let's do that.
  
  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 .
  
  Now: for what values of x do you think that this will converge?
  
  Note that this is a variation on the theme of the ratio test: we don't consider the power terms, but only the coefficients.
- Examples:
  - Consider :
    1. What does Theorem 2 say will be the radius of convergence of the series for this series?
    2. What about the endpoints?
    3. What is the interval of convergence -- the interval around the center of the power series for which the series converges -- in this case?
    4. How well do the truncated partial sum functions approximate ${F(x)=\frac{1}{1-x}}$ ?
  - #7, p. 769
  - #19
  - #22

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