# Wednesday 29 November class

Apologies for the delayed posts while I was getting my computer fixed… I’m working on catching up now

Topics:

• Discussion of finding intervals of convergence for power series. The solutions to the quiz are posted here.

• Important Taylor series that you should recognize when you see them. See the list below.

• Using known Taylor series to get new Taylor series, by multiplying by a polynomial or substituting a polynomial in place of x.

• New topic: finding the area between two curves (just an introduction).

These are true areas, not the kind of “area” we get from the definite integral, which could be positive or negative. Therefore we have to make sure and look at the graph to get the subtraction in the correct order, so the result will be positive (and no piece of it will be negative).

Important Taylor series: (see more in Key Idea 32 on p.482, but I’ve changed the list a bit)

Make sure that you also know or can easily figure out the intervals of convergence!

$e^{x} = \displaystyle \sum_{n=0}^{\infty}\frac{x^{n}}{n!}$

$\sin(x) = \displaystyle \sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n+1}}{(2n+1)!}$

$\cos(x) = \displaystyle \sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n}}{(2n)!}$

$\ln(x) = \displaystyle \sum_{n=1}^{\infty}\frac{(-1)^{n+1}(x-1)^{n}}{n}$

$\ln(1+x) = \displaystyle \sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^{n}}{n}$ * not in textbook

$\frac{1}{1+x} = \displaystyle \sum_{n=0}^{\infty}x^{n}$ (Geometric series)

Apologies for the “Formula does not parse” error below: the formula is correct, but WordPress is not interpreting it for some reason. Please look for the formula for the inverse tangent in the textbook (or elsewhere).



Homework:

• Review the examples discussed in class. A good way to familiarize yourself with the important Taylor series is to find them from scratch a few times. The arctangent formula is probably the most difficult (because you must repeatedly use the quotient rule).

• Finish the WeBWorK on Taylor series, and try a little of the areas between curves. Please do not wait to the last minute! In case the problem asks just for the area bounded between two curves, but does not tell you where to start and end the integral, you have to figure out where the two curves intersect. Using Desmos may help with this.

• There will be a quiz next time: the topic will be using known Taylor series to give new ones, and finding intervals of convergence for Taylor series.

• Don’t forget that Test 4 is scheduled for next Wednesday. The review self-tests will be in a separate post. This will be a shorter test than usual, so I will add to it a “makeup” problem similar to one that was commonly missed on a previous test. Make sure you look for it. This will give you an opportunity to improve a previous test score.

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!