In this video we are going to take a look at Taylor Series and Maclaurin Series. Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. If the Taylor series is centered at zero, then that series is also called a Maclaurin series. So the Maclaurin series is just a special case of Taylor series. We are also going to talk about Taylor Polynomials. The formula for the Taylor Polynomial looks exactly the same for the Taylor series, except what we do is instead of running it to infinity, we cut the thing off at the degree k term.
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.
*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.
Taylor Series and Maclaurin Series
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