In this lecture Professor Murray will teach you the Taylor Series and Maclaurin Series. After an in depth analysis of the two series, you will learn the Taylor Polynomial and finish off with several examples of each.
Definitions: The Taylor
Series for a function f (x) around a center value a
is the power series
(a) represents the n-th derivative of f, with a
The Maclaurin Series for f
(x) is just the special case of the Taylor Series around the
center value a = 0:
The Taylor polynomial is what
you get when you cut off the Taylor Series at the degree k
Hints and tips:
In many cases, you do not want to
use the formulas above to find the Taylor Series of a function,
because the derivatives get too messy. Instead, start with some
known Taylor Series for some common function and derive other series
from the known series using the following techniques:
Algebraic manipulations, e.g. multiplying by x.
Substitutions, e.g. replacing x by 2x or x² .
Derivatives and integrals.
Multiplying or dividing two series together.
You should memorize the Maclaurin
Series for ex , sin x, and cos x
at the very least, and probably for 1/(1−x), arctan x,
and ln(1 − x) as well.
Sometimes you cannot find the
general pattern for a Taylor Series, especially those that are not
centered at a = 0. However, you can still find the first
few terms, and this might be enough for computations.
The Taylor series for a polynomial
is just the polynomial itself. A common mistake is to think that the
Taylor polynomial Tk (x) has k
terms. k refers to the degree, not the number of terms. So,
for example, the Taylor polynomial T4
(x) for f (x) = sin x centered around a
= 0 is T4
(x) = x − x³⁄ 6 , because the term
of x4 is zero.
Taylor Series and Maclaurin Series
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.