A student claims that, for any analytic function g centered at the origin, increasing the degree of a Taylor polynomial necessarily reduces the maximum possible error of the approximation on a fixed interval around the center. The student offers no calculations but argues that "higher–degree terms make the approximation better."
a. Provide a clear, concise explanation assessing the student’s claim. Your response should reference appropriate mathematical properties that justify whether the claim is always true, conditionally true, or false. Do not perform explicit error calculations; instead, reason qualitatively about how the size of the (n+1)st derivative and the rapid growth of factorial terms influence the magnitude of the remainder.