Taylor Series
Taylor series
, or Taylor polynomial
is a series that can REPRESENT a function, regardless what function it is.
▼Refer to 3Blue1Brown for animation & intuition: Taylor series | Chapter 10, Essence of calculus
"Taylor Series is one of the most powerful tools Math has to offer for approximating functions." - 3Blue1Brown
▲ Notice: The
Taylor Series
is aPower Series
, which means we can use a lot of techniques of power series on this to operate it easily.
The main purpose of using a Taylor Polynomial
is to REPLACE the original function with a polynomial, which it is easy to work with.
More importantly, by adding more & more terms into the polynomial, we can approximate the function more precisely:
►Refer to joseferrer: Mathematical explanation - Taylor series ►For More animation, visit Desmos: Taylor Series Visualization
Example
First to know the formula of
Taylor Series
centred atx=1
:The problem is asking the coefficient of
(x-1)³
, means all the rest part in the formula, which is:And it also means the
n=3
, so the coefficient becomes:Let's evaluate the
f'''(1)
:So the coefficient is:
Example
Let's express the Taylor polynomial to the
nth degree
as:Since it's asking for the series to the
3rd degree
, then it becomes:And we only need to find out every degree of derivatives, and we will get:
So the Taylor polynomial then is:
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