TITLE: Lagrange remainder vs. Alternating Series Estimation Theorem: do they always give you the same error bound?
QUESTION [2 upvotes]: Given a function and its nth degree Taylor series approximation, we can use the Lagrange form of the remainder to get a maximum value of the error of approximation. If the series is also an alternating series, we can also use the Alternating Series Estimation Theorem to get a maximum value of the error of approximation. Will the two maximums always be the same?
I have tested a few cases and they turn out to be the same. I'm looking for a proof that they are always the same, or else a counter-example.

REPLY [0 votes]: Alright - how's this for a generalization?  When a Taylor polynomial expansion P(x) for function f(x) happens to alternate in signs, then both the Alternating Series Estimation Theorem and the Lagrange form of the remainder provide us with upper bound errors between the P(x) and f(x).  However, the Alternating Series remainder will always be less than or equal to the Lagrange remainder and therefore will give us a better idea of the accuracy of our estimation.
{"set_name": "stack_exchange", "score": 2, "question_id": 902475}