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Finding Taylor or Maclaurin Series for a Function

Samuel Baker

Samuel Baker

8 min read

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Study Guide Overview

This study guide covers Taylor and Maclaurin series. It explains how to find the Taylor series representation of a function, including the general formula and the special case of Maclaurin series (centered at x = 0). Important Maclaurin series for common functions like , sin(x), cos(x), and 1/(1-x) are listed. Finally, the guide provides practice problems demonstrating how to find Taylor series representations and list specific terms for given functions and centers.

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. 🎸

!Untitled

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:

n=0f(n)(a)n!(xa)n=f(a)+f(a)(xa)+f(a)2!(xa)2+f(a)3!(xa)3+...+f(n)(a)n!(xa)n\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)f^{(n)}(a) is the nthn^{\text{th}} deriviative of the function and f(0)(a)=f(x)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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Question 1 of 9

What is the center of a Maclaurin series? 🤔

x = 1

x = -1

x = 0

Any real number