Finding Taylor or Maclaurin Series for a Function

#10.14 Finding Taylor or Maclaurin Series for a Function

Taylor who now? And no, we’re not talking about the iconic artist that has reached global audiences shown below. 🎸

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Taylor Swift

Image courtesy of Wikimedia Commons

#🤔 What’s a Taylor Series?

Even so, Taylor series are as iconic as Taylor Swift in a sense that they combine the following ideas:

1. From [10.11] We can approximate functions as polynomials using the Taylor approximations theorem.
2. From [10.13] We can represent functions as power series, which is made up of a sequence and a real number serving as its center.

Taylor Series: For a function f(x), its Taylor series approximation at x = a is:

$$\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}\cdot(x-a)^n=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+...+\frac{f^{(n)}(a)}{n!}(x-a)^n$$

Where $f^{(n)}(a)$ is the $n^{\text{th}}$ deriviative of the function and $f^{(0)}(a)=f(x)$.

A Taylor series, essentially, 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. It allows us to approximate functions and calculate their values at different points. ✅

You might also come across a series called Maclaurin series. If you do, don’t be scared! In fact, Taylor series centered at x = 0 are so common that they have a special name called the Maclaurin series.

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