Finding Taylor or Maclaurin Series for a Function

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 eˣ, 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:
- From [10.11] We can approximate functions as polynomials using the Taylor approximations theorem.
- 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:
Where is the deriviative of the function and .
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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