The purpose of Taylor errors is to quantify the difference between a given function and its Taylor polynomial approximation. This allows us to estimate the accuracy of our approximation and make informed decisions about the level of precision needed for a particular calculation.
Taylor errors are typically calculated using the remainder term in the Taylor series expansion of a function. This remainder term takes into account the higher-order derivatives of the function and the distance between the point of approximation and the point of evaluation.
Yes, Taylor errors can be negative. This indicates that the Taylor polynomial overestimates the value of the function at a given point. However, in most cases, Taylor errors are positive, meaning that the Taylor polynomial underestimates the function.
Taylor errors are closely related to the convergence of a series. In fact, the remainder term in the Taylor series expansion can be used to test for the convergence of a series. If the remainder term approaches zero as the number of terms in the series increases, then the series is said to converge.
It is not possible to completely eliminate Taylor errors, as they are inherent in the use of Taylor polynomials to approximate a function. However, the accuracy of the approximation can be improved by using a higher-order Taylor polynomial or by reducing the distance between the point of approximation and the point of evaluation.