6.3 Taylor and Maclaurin Series (p.561–562, 573–576)
6.4 Working with Taylor Series (p.584–587, 590–592)

P. 578: 118-123 all, 140—147 all, 151—155 all
P. 596: 203, 206, 207, 209, 219–223 odd

Webwork Taylor series due Tuesday, May 5

Notes from lecture and office hours: https://www.dropbox.com/s/f8gxpteui7easip/Note%20Apr%2029%2C%202020.pdf?dl=0

Motivation

You already know the motivation for this topic: Taylor polynomials approximate a function near a point…the higher the degree, the better the approximation…the best approximation would be an “infinite-degree polynomial” but we don’t have such things, we have power series instead.

So a Taylor series is just the power series you would get if your Taylor polynomial could go on forever!

Taylor (and Maclaurin) series

Since a Taylor series for a function $f(x)$ centered at $x=c$ is a power series, it has an interval of convergence, you can ask what does the Taylor series converge to on that interval? This is actually a stupid question because of what we already know: Taylor polynomials approximate a $f(x)$ near $x=c$ and the higher the degree, the better the approximation. This just means that on the interval of convergence, the Taylor series of $f(x)$ converges to….$f(x)$!

This is basically the whole punchline of Chapters 5 and 6, but we spoiled it because we said from the beginning that we wanted the “Taylor polynomial that best approximates $f(x)$ near $x=c$.

If the center $c=0$ then the Taylor series is sometimes called the Maclaurin series (this was the case for Taylor and Maclaurin polynomials too).

This video (6 minutes) shows how to find the Taylor series for $f(x)=e^x$ centered at $x=1$.

This video (6 minutes; start at 1:00) shows how to find the Taylor/Maclaurin series for $f(x)=\cos(x)$ at $x=0$.

One you already know the Taylor series for $f(x)= \cos(x)$ at $x=0$, you can use this knowledge to build Taylor series of more complicated functions at $x=0$. This video (8 minutes) shows how to find the Taylor/Maclaurin series for $f(x)= x^3 \cos(x^2)$ without having to take a bunch of derivatives.

Interval/radius of convergence

Finding the interval or radius of convergence for a Taylor series is no different than finding  the interval or radius of convergence for a Taylor series for any old power series:

  • use the ratio test to set up an inequality to determine the radius of convergence,
  • use the center and radius of convergence to determine the largest open interval on which the power series converges,
  • test the convergence of the series at the two interval endpoints separately.