- Homework 11.3 graded:
- 1
- 22
- Your homework for 11.4 is due today. Questions?
- The quiz opportunity is over 11.8: power series
- First, a definition:
- Series that converge absolutely, and
- Series that don't.
- Examples:
- #5, p. 761
- #12, p. 761
- It turns out that conditionally convergent series can have their terms rearranged to converge to any limit you like (let's take a look at "Rearrangements", on p. 761). Hmmm.... Is this the end of commutativity?!
- The Ratio and Root Tests
- There are two main tests for convergence of series in this section:
This result says that eventually the ratio of successive terms is effectively constant,
: the terms of the sequence approach a "common ratio" as
. What kind of series looks like that? A geometric series:
Examples:
- #2, p. 761
- #3, p. 761
The ratio test is effectively a self-referrential comparison test: we compare terms of
with other terms
(rather than with some other series).
This result says that eventually the absolute values of the terms are effectively equal to
: what kind of series looks like that? A geometric series!
No wonder the results of the tests look exactly the same.... Too bad we didn't just use the same letter for the limits in both cases!
Notice that, once again, limits of sequences plays an important role! Series are just sums of sequences, after all; we're focused on how terms behave (root test), how successive terms behave (ratio test), or how partial sums behave.
Examples:
- #7, p. 761
- #13, p. 761
- There are two main tests for convergence of series in this section:
- 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
,
Now what did we assert about
For what values of x would this converge?
Apply the ratio test for arbitrary x, and what do you discover?
So
Let's look at an example where our radius of convergence is not so nice: consider
. One can show that
Now: for what values of x do you think that this will converge? And why does it make sense?
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
- What about the endpoints?
- #7, p. 769
- #19
- #27
- What does Theorem 2 say will be the radius of
convergence of the series for
- 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.