6.1 Power Series and Functions (p.531–537)
6.2 Properties of Power Series (p.544–548, 552–557)

P. 541: 13-21 odd, 24, 28
P. 558: 87—90 all, 96, 97

Webwork: Power Series due Sunday, May 3

Lecture notes:

https://www.dropbox.com/s/i86orzrwnoc15s8/Note%20Apr%2027%2C%202020.pdf?dl=0

Sorry, no video again today! I’ll see if I can figure Webwex out…

These two sections are scheduled for one day, because they’re really about different aspects of one topic: power series.

Remember that before we discussed sequences and series, we discussed Taylor Polynomials. Higher and higher degree Taylor polynomials approximate a function better and better near a specified point (the “center”). So the “best approximation” would be an “infinite degree polynomial.” Except infinite degree polynomials don’t exist: those are called power series.

Since we’ve finished discussing the convergence or divergence of series where we’re adding up infinitely many numbers, we’re ready to start discussing the convergence or divergence of series where we’re adding up infinitely many powers of $x$. That’s what a power series is.

Motivating example

You’ve actually seen your first examples of power series already; take a look at the Webwork set Intro to Series problem #14. One version of the series in that question looks like:

$\sum_{n=1}^\infty \frac{x^n}{2^n}.$

It’s not really a geometric series, because of that $x$, but it’s almost like a geometric series where the common ratio $r$ is not a number but is $\fra{x}{2}$. Whether the series converges or diverges depends on whatever value $x$ is. When you substitute a value for $x$, you’re left with a series of numbers, like you were studying in Chapter 5. One easy value to check is when $x=0$. In this case the series looks like:

$\sum_{n=1}^\infty \frac{0^n}{2^n} = 0 + 0 + 0 + \dots = 0$

So for $x=0$, the series $\sum_{n=1}^\infty \frac{x^n}{2^n}$ converges to 0. In this example, $x=0$ is the center of the power series. A power series always converges at its center.

But you don’t want to use trial and error to find the values of $x$ for which the power series converges. In that Webwork problem, you were able to find these values of $x$ by treating $\frac{x}{r}$ as $r$ and solving $|\fra{x}{2}| <1$, using what you know about convergent geometric series.

In this example, $x$ must be in the open interval $(-2,2)$. That’s called the interval of convergence for this power series. Notice that the center $x=0$ is right in the center of the interval. We say that the radius of convergence for this power series is 2, because that’s the maximum distance $x$ can be from the center for which the series will converge.

You can see the first 10 partial sums of this series (in rainbow colors) and the interval of convergence (shaded in grey) in this Desmos graph. You can see that inside the shaded region, the graphs look like they’re converging to something, but outside the shaded region, they’re just doing whatever, so they’re diverging there.

You can also see the graph of $f(x) = \frac{\frac{x}{2}}{1 – \frac{x}{2}}$ in black. I didn’t simplify this expression because I wanted to emphasize: when a geometric series converges, it converges to $\frac{a}{1-r}$. The series $\sum_{n=1}^\infty \frac{x^n}{2^n}$ behaves like a geometric series where both $a$ and $r$ are $\frac{x}{2}$. So inside the interval of convergence $(-2,2)$ the series $\sum_{n=1}^\infty \frac{x^n}{2^n}$ converges and we know what it converges to: $f(x) = \frac{\frac{x}{2}}{1 – \frac{x}{2}}$.

Other examples

Typically, you will be asked to determine the interval of convergence and radius of convergence  of a power series, and not what the power series converges to on the interval of convergence. This motivating example was practically geometric (except for that $x$), which is what made it easy to determine what the power series converges to. Power series usually start at $n=0$ or $n=1$ but don’t be worried if they start at some other value of $n$.

Remember that the ratio test is handy for series that are “eventually practically geometric,” so that’s the test you’ll be using for other power series. Don’t forget, if the value of the limit

$\rho = \lim_{n \to \infty} \abs{\frac{a_{n+1}}{a_n}}$

is 1, then the ratio test is inconclusive. You’ll find that this is what happens at the endpoints of the interval of convergence, so you’ll have to determine the convergence/divergence of those two series separately, using the appropriate convergence test.

The following videos show examples that will help you with Webwork Power Series EXEPT for #11, 12, 13. We’ll get to those in a minute.

As usual, this account has way more amazing examples (search for “Power Series”).

Expressing functions as power series

In Webwork Power Series  #11, 12, 13, you’re asked to represent different functions as power series. You’ll learn how to do this using Taylor Series on Wednesday, but that’s not the approach you’re supposed to use here.

Usually when we’re dividing polynomials using long division, the polynomial we’re dividing by has a lower degree than the polynomial we’re dividing. (So to find $\frac{p(x)}{q(x)}$, the degree of $p$ would be greater than the degree of $q$.) But this doesn’t have to be the case.

A good exercise would be to use long division to see the following fact:

$f(x) = \frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots = \sum_{n=0}^\infty x^n.$

That is, the function $f(x) = \frac{1}{1-x}$ may be represented by the power series $\sum_{n=0}^\infty x^n$ on its interval of convergence.  (Notice this power series starts at $n=0$).

Once you know this fact, you can use it to represent other functions using power series.

For example, say we want to represent $g(x) = \frac{1}{1+x}$ as a power series. This is not the same function as $f(x)$ above, but we can replace $x$ by $-x$ in the expression for $g(x)$ and then we get something that looks like the same form of $f(x)$. Then we just replace $x$ by $-x$ on the right-hand side of the above fact:

$g(x) = \frac{1}{1+x} = \frac{1}{1-(-x)} = 1 + (-x) + (-x)^2 + (-x)^3 + \dots = \sum_{n=0}^\infty (-x)^n.$

  • This video (6 minutes) connects the above fact with what you know about geometric series to express another function as a power series. This is similar to your Webwork problems.
  • Even when functions are not given in the form where you can apply the fact directly, you can sometimes manipulate the expression algebraically so that you can apply the fact as in this video (7 minutes) or in this video (9 minutes).