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

►Refer to Khan academy: Taylor & Maclaurin polynomials intro (part 1)
▼Refer to xaktly: Taylor Series

(▲ C represents the centre where we're centred at to approximate the function.)

▲ Notice: The Taylor Series is a Power Series, which means we can use a lot of techniques of power series on this to operate it easily.

We could expand it and make it clearer ▼:

The main purpose of using a Taylor Polynomial is to REPLACE the original function with a polynomial, which it is easy to work with.

etc., we can express the function f(x) = eˣ as ▼:

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

Solve:

• First to know the formula of Taylor Series centred at x=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

Solve:

• 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: