Finding the Maximum Remainder in a Taylor Series: Explained

In summary, the conversation discusses the thought process behind finding the maximum remainder of a Taylor series. The speaker mentions reading a Wikipedia article and their confusion with a diagram. Another person explains that Taylor polynomials are used to approximate a function, and the accuracy can be adjusted by increasing the order. The conversation also mentions the importance of accuracy for engineering and other fields.
  • #1
Skrew
131
0
Hello, I was wondering if anyone could explain to me the thought process behind how you find the maximum remainder of a Taylor series?

I read the wiki article and didn't help me at all,

http://en.wikipedia.org/wiki/Taylor's_theorem

My book talks about something like this(image is wiki's):

59016b56cc025694b4e3baf84adf71c1.png


but I don't understand how its derived or what it really means.

Thanks for any help.
 
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  • #2
We never really did too much with this, however in Taylor polynomials you are generating an approximation through a polynomial. How accurate you are is dependent upon how many time you can differentiate the function and what value you choose for the function. The polynomial approximates the function as the order, n, approaches infinity however computing this isn't really possible so when you are trying to find the value of a function at point you really are asking yourself how accurate do you wish to be. By adjusting n in the inequality you can attempt to increase accuracy. This is important for engineering and other experts when they demand a certain tolerance of error.
 
  • #3
Basically that fancy equation tells you the term after your last time is your error.
 

Related to Finding the Maximum Remainder in a Taylor Series: Explained

What is a Taylor Series Remainder?

A Taylor Series Remainder is the difference between the actual value of a function and its approximation using a Taylor polynomial. It represents the error or remainder term in the Taylor series expansion.

Why is the Taylor Series Remainder important?

The Taylor Series Remainder is important because it allows us to approximate a function with a polynomial and determine the accuracy of the approximation. It also helps in evaluating complicated functions and finding the best polynomial approximation for a given function.

How is the Taylor Series Remainder calculated?

The Taylor Series Remainder is calculated using the Taylor series expansion formula, which involves taking the nth derivative of the function and evaluating it at a specific point. It is then multiplied by the remainder term, which is the value of x raised to the (n+1)th power divided by (n+1)!.

What is the significance of the remainder term in Taylor series?

The remainder term in Taylor series is significant because it allows us to determine the accuracy of the Taylor polynomial approximation. As the degree of the polynomial increases, the remainder term decreases, indicating a more accurate approximation of the function.

How does the Taylor Series Remainder relate to the convergence of a Taylor series?

The Taylor Series Remainder is directly related to the convergence of a Taylor series. If the remainder term approaches zero as the degree of the polynomial increases, then the Taylor series converges to the actual value of the function. If the remainder term does not approach zero, then the Taylor series does not converge and the approximation is not accurate.

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