**(1)** What are the main differences, in terms of "usefulness" while solving problems (even at research level), among [Cauchy][1], [Lagrange][3], and [Schlömilch][2]'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?

> 1. BLUMENTHAL, L. M., *Concerning the Remainder Term in Taylor's Formula*.
> Amer. Math. Monthly **33**, pp. 424-426, 1926.

> 2. BEESACK, P. R., *A General Form of the Remainder in Taylor's Theorem*.
> Amer. Math. Monthly **73**, pp. 64-67, 1966

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Note: Similar questions were posed on M.SE, but no satisfactory answer was provided (even after that 2 bounties expired).

  [1]: http://mathworld.wolfram.com/CauchyRemainder.html
  [2]: http://mathworld.wolfram.com/SchloemilchRemainder.html
  [3]: http://mathworld.wolfram.com/LagrangeRemainder.html