So now we’re ready to take our power function skills to the next level!
The Taylor Series, sometimes referred to as the Taylor Polynomial, allows us to find a polynomial function that approximates another function by first choosing a number (center) both the function and the polynomial share.
As seen in the introductory video for Taylor and Macluarin Series, the overall goal is to take a “hard” function and turn it into something easier.
For example, when we were learning how to represent power functions we first had to recognize the similarities it had to the geometric series? And then we applied term-by-term differentiation or integration to create the power series representation for the given function.
But this technique has its limitations. As Paul’s Online Notes accurately states, that while there are many functions that can be related to the geometric series, there are even more that cannot.
Thankfully, the Taylor Series will give a general rule for generating power series for functions that have derivatives of all orders! No limitations!
Sweet!
So we will begin with the general formula and techniques for writing a Taylor Series. We will use tables to keep us organized and also tricks for simplifying terms.
Next, we will expand our knowledge by examining the five most important Taylor/Maclaurin Expansion formulas.
Important Taylor and Maclaurin Series
A Maclaurin Series is a Taylor Series centered at zero. And by knowing these basic rules and formulas, we can learn to use them in generating other functions as well as how to apply them to Taylor Series that are not centered at zero.
These expansion formulas will help us to write polynomials quickly, and how they empower us to simplify more complex problems, such as learning how to evaluate, differentiate, and Integrate a Taylor Series Polynomial.
Taylor Series Video
Taylor Series Examples
Taylor Series Overview with Examples in Calculus
- Taylor Series Overview
- Example #1 for writing a Taylor Polynomial for a given function
- Example #2 for writing a Taylor Polynomial for a given function
- Example #3 for writing a Taylor Polynomial for a given function
- Example #4 for writing a Taylor Polynomial for a given function
- Five Basic Maclaurin/Taylor Expansion Formulas
- Examples using Macluarin Expansions
- More Examples using Maclaurin Expansions
- Power Series and Taylor/Maclaurin Expansion Examples
- Example #1 using Taylor Expansions to simplify functions and find general terms
- Example #2 using Taylor Expansions to simplify functions and find general terms
- Example #3 using Taylor Expansions to simplify functions and find general terms
- Example of how to evaluate a function using a Taylor Series
- Example #1 of how to Integrate a function using a Taylor Series
- Example #2 of how to Integrate a function using a Taylor Series
- Example of how to find a value of a Taylor Polynomial
Get more examples and over 450 HD videos with your subscription
Monthly and Yearly Plans Available
Still wondering if CalcWorkshop is right for you?
Take a Tour and find out how a membership can take the struggle out of learning math.