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(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?

  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


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

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    $\begingroup$ A link to the companion question at M.SE: math.stackexchange.com/q/1059461/166535 $\endgroup$ Dec 31, 2014 at 18:28
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    $\begingroup$ When i studied at bachelor at Kiev, the lectures in Analysis went along the book of Dorogovtsev "Mathematical Analysis", and i saw there those formulae mentioned in the "papers" you cite. I always wondered how much pedantic Dorogovtsev must have been, and (the book is written almost only in maths symbols) felt quite unwell reading his masterpiece. The fruitful thing is that now my stomach is much stronger than when i was at school! $\endgroup$
    – Victor
    Jan 1, 2015 at 8:02

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