Finding the Maximum Remainder in a Taylor Series: Explained

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SUMMARY

The discussion centers on understanding the maximum remainder in a Taylor series, specifically how to derive and interpret it. The maximum remainder is determined by the order of the Taylor polynomial and the function's differentiability. As the order, n, approaches infinity, the polynomial increasingly approximates the function, but practical computation limits this. Adjusting n in the remainder inequality allows for control over the accuracy of the approximation, which is crucial for engineering applications requiring specific error tolerances.

PREREQUISITES
  • Understanding of Taylor series and Taylor polynomials
  • Knowledge of calculus, specifically differentiation
  • Familiarity with error analysis in numerical methods
  • Basic concepts of polynomial approximation
NEXT STEPS
  • Study Taylor's theorem and its implications in approximation theory
  • Explore error bounds in Taylor series, focusing on Lagrange's form of the remainder
  • Learn about numerical methods for function approximation
  • Investigate applications of Taylor series in engineering and physics
USEFUL FOR

Students of calculus, mathematicians, engineers, and anyone involved in numerical analysis or approximation methods will benefit from this discussion.

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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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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.
 
Basically that fancy equation tells you the term after your last time is your error.
 

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