(1) What are the main differences, in terms of "usefulness" while solving problems (even at research level), among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula? Could you provide examples of situations where one form "works" better than another?
(2) Also, can you show instances in which the generalizations proposed in the following articles are fruitful?
- BLUMENTHAL, L. M., Concerning the Remainder Term in Taylor's Formula. Amer. Math. Monthly 33, pp. 424-426, 1926.
- BEESACK, P. R., A General Form of the Remainder in Taylor's Theorem. Amer. Math. Monthly 73, pp. 64-67, 1966
Note: Similar questions were posed on M.SE, but no satisfactory answer was provided (even after that 2 bounties expired).