Good Question 16

I had an email last week from a teacher asking, how come I can use a substitution to find a power series for  , and for  , but not for  ? The answer is that you can. Substituting (2x) in to the cosine’s series give you a Taylor series centered at x = 0, a…

Introducing Power Series

The posts for the next several weeks will be on topics tested only on the BC Calculus exams. Continuing with some posts on introducing power series (the Taylor and Maclaurin series) Introducing Power Series 1 Two examples to lead off with. Introducing Power Series 2 Looking at the graph of a power series foreshadows the…

Synthetic Summer Fun

Today, for some summer fun, let’s look at synthetic division a/k/a synthetic substitution. I’ll assume you all know how to do that since it is a pretty common pre-calculus topic and even comes up again in calculus. Why Does Synthetic Division Work? An example: consider the polynomial . This can be written in nested form…

Sequences and Series

AP Type Question 10 Sequences and Series – for BC only Convergence tests for series appear on both sections of the BC Calculus exam. In the multiple-choice section students may be asked to say if a sequence or series converges or which of several series converge. The Ratio test is used most often to determine…

Inrtoducing Power Series 2

In our last post we found that we could produce better and better polynomial approximations to a function. That is, we produced a set of polynomials of increasing degree that had the same value for the functions and its derivatives at a given point. To see what is going on I suggest we graph these approximating polynomials…