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.