Taylor series error term - graphical representation

In summary, the conversation discusses the use of Taylor series to represent error bounds in a graphical way. The link provided shows an example of a second order error bound, and the question is raised if it is possible to represent higher order error bounds in a similar manner. The concept of the Mean Value Theorem is mentioned, and a resource for further information is provided.
  • #1
wizkhal
6
0
Hello all,
Recently I've found something very interesting concerning Taylor series.
It's a graphical representation of a second order error bound of the series.
Here is the link: http://www.karlscalculus.org/l8_4-1.html [Broken]

My question is: is it possible to represent higher order error bounds in a similar way?
For example: third order error term would have "3! = 6" in a denominator...
I know that Taylor series is based on Mean Value Theorem and I know the proof of it.
However it would become much clearer if it was possible to represent error bounds in a graphical way.

Have a nice weekend.
 
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  • #2
Yes, see any calculus book, or this

http://mathworld.wolfram.com/SchloemilchRemainder.html

It follows from the mean value theorem

Often in simple examples the error is well approximated by the next term as in

[tex]\sin(x) \sim \sum_{k=0}^\infty \frac{x^k}{k!} \sin\left(
x+k \frac{\pi}{2}\right)[/tex]
 
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What is a Taylor series error term?

A Taylor series error term refers to the difference between the actual value of a function and the value predicted by using a finite number of terms from its Taylor series. It helps to quantify the accuracy of the approximation of a function using its Taylor series.

How is the Taylor series error term represented graphically?

The Taylor series error term is typically represented graphically as a function of the number of terms used in the approximation. This can be shown as a line graph, with the x-axis representing the number of terms and the y-axis representing the error term.

What does a small Taylor series error term indicate?

A small Taylor series error term indicates that the approximation using the given number of terms is very close to the actual value of the function. This means that the Taylor series is a good representation of the function and can be used to accurately estimate its values.

Can the Taylor series error term be negative?

Yes, the Taylor series error term can be negative. This indicates that the approximation using the given number of terms underestimates the actual value of the function. However, as the number of terms increases, the error term becomes smaller and eventually becomes positive, indicating that the approximation is now overestimating the actual value.

How does the Taylor series error term change as the number of terms increases?

As the number of terms in the Taylor series increases, the error term becomes smaller, indicating a better approximation of the function. However, there may be a point where adding more terms does not significantly decrease the error term, and it may even start to increase. This is known as the convergence point of the Taylor series.

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