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Issac G.

asked • 02/12/16

How does the Lagrange Form of the Error Bound give you the maximum expected error?

The Lagrange form of the Error Bound states that the n+1 term of a Taylor Approximation gives you the remainder such that F(x)=Pn(x)+Rn(x), but what does that mean? I understand that the goal is to find the maximum error that can be expected, but the process does not seem logical. Why is it the next term that gives you the error? Why cant it be the one after? Also, when attempting to evaluate M such that it is greater than or equal to the n+1 derivative of F(z) where z is some value between c (where the function is centered) and x, how does finding the largest value for z in the interval give you the maximum value? I see that the biggest value is obviously going to maximize the expected error, but I don't see how evaluating the n+1 derivative of the function at z correlates with the Taylor series since all the derivatives beforehand were evaluated at c.

1 Expert Answer

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Arnold F. answered • 02/12/16

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Issac G.

When adding the Remainder to the Taylor Approximation how does it give you the exact function? What is the Remainder representing?
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02/13/16

Arnold F.

Rn(x) is a function that represents the difference between the approximation and the exact value of f(x). If we know the exact value of the remainder (we don't) we would know the exact value of f(x). But if we can calculate the maximum possible value of R then that can be used as the maximum possible error incurred when using the Taylor approximation for f(x).
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02/13/16

Issac G.

Thank you! That makes a lot more sense. However, how come we use the n+1 and not the n+2 or n+3 derivative?
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02/15/16

Arnold F.

We don't use a higher derivative because then there would be a "discrepancy" between the approximation (which stops at nth derivative) and the maximum error (which you would be calculating using a higher derivative.) 
 
The discrepancy is essentially the n+1 term since it is not part of the approximation and not part of the remainder.
 
Look at this web page and look at lines 16-18:
http://mathworld.wolfram.com/TaylorSeries.html
 
 
Let me know if that helps.
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02/15/16

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