Monday 27 November class


• Discussion of some problems finding radius and interval of convergence for power series.

Note: it is important to be aware of which test you are using (so that you carry it out correctly) and that you write your work in a logical way. Know what you are doing and what it is supposed to accomplish at each step.

Also we are using the fact that |x-c| < R translates into -R < x-c < R. There are other ways to handle absolute value inequalities, but since this is the only type of absolute value inequality we will need in this class and we want to get the interval of convergence, this is the best and most direct way to do it.


• Taylor series (introduction)

A Taylor series is just an infinite Taylor polynomial. Notice that this means that the function must have infinitely many derivatives, which is not always the case. (This may limit the interval of convergence.)

A taylor series being a power series, it will have a radius and interval of convergence. There is also the question of exactly what it converges to, if the series converges. There is a way to prove that the Taylor series converges to the original function, using the remainder theorem for Taylor polynomials, which works in many cases.

We will assume that for the Taylor series we discuss in this class, they converge to the original function inside the radius of convergence. (Problems can happen at the endpoints of the interval of convergence.)

The simplest and maybe most beautiful example of a Taylor series is the Maclaurin series for e^{x}, which we found in class

1 + x + frac{x^{2}}{2} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \dots = \displaystyle\sum_{n=0}^{\infty}\frac{x^{n}}{n!}

We showed using the Ratio Test that this converges on the whole real line. (It is also possible to prove that what it converges to is the function e^{x})


Another example: the Taylor series for \ln(x) centered at x=2. (Note that \ln(x) does not have a Taylor series around x=0, since \ln(0) is undefined. So any Taylor series for \ln(x) must be centered around some x-value where the natural logarithm is defined.)

This example is worked out in Paul’s Online notes.(Example 7)  It turns out to give this:

\ln(2) + \displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}{n2^{n}}(x-2)^{n}

There are two more commonly used Taylor Series for the natural logarithm, which are Maclaurin series for \ln(1-x) and \ln(1+x). Using the series for \ln(1+x) we can prove that the alternating harmonic series converges to \ln(2), by substituting x=1 into that series. (We should first prove that the interval of convergence includes x=1, of course!)

There are important Taylor series that are commonly used and that you should become familiar with. We will make a list of some of them next time.



• Review the examples discussed in class. Make sure that you understand how we are implementing the various tests for convergence at each step.

• Review how to find Taylor polynomials if you need to do so.

There is a series of videos and practice problems on Khan Academy if you are interested. (The iink is to the first video in the series.)

• Do the WeBWorK: please do not wait to the last minute! The WeBWorK on Power Series is due Tuesday evening. The WeBWorK on Taylor series is not due until Sunday, but again, please do not wait, and make sure to post questions on Piazza if you encounter difficulty.

• I have posted two problems on Piazza which may be done (in Piazza) for extra credit. These are intended to be used ONLY by people who have  not yet had a chance to put a problem on the board in class: please respect this, and also please only do ONE problem if you choose to do so. I may post more problems from time to time, so make sure you are getting alerts from Piazza. (On a computer, you can go into your account/email settings and check how you are being notified and make changes, if necessary.)

• It would be a good idea to fill out the ThinkingStrategicallyPreTestSurvey at this point to start preparing for Test 4!


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!