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While solving problems and exercises, so far I've only used Lagrange's form of the remainder.
Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's formula.

So my question is:

Could you show some examples of exercises where Cauchy's (or other) forms of the remainder come in handy for some reason? Could Lagrange's form be nevertheless applied in such cases to solve the problems? Is there a rule of thumb to decide which form is better to use in a given occasion?

Dal
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    Here's one. – David Mitra Aug 12 '14 at 13:11
  • Dear @DavidMitra, could you show other examples too? – Dal Dec 16 '14 at 16:39
  • @Dal That's the only one I know about. But, I'll think about it. – David Mitra Dec 16 '14 at 16:42
  • @DavidMitra, thanks! – Dal Dec 16 '14 at 16:42
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    It does not answer the question, but you may also be interested by the Schlömilch remainder. Here is a note about the different remainders and their derivation (in french). The french Wikipedia article about Taylor's theorem has also some information about these (and also the so-called Taylor-Young and Taylor-Laplace). – Jean-Claude Arbaut Dec 16 '14 at 16:43
  • @Jean-ClaudeArbaut Thank you for the references! :) I knew Schlömilch remainder (indeed, I had asked this question http://math.stackexchange.com/questions/1059461/differences-among-cauchy-lagrange-and-schl%C3%B6milch-remainder-in-taylors-formula). – Dal Dec 16 '14 at 16:51
  • @Dal Indeed :-) – Jean-Claude Arbaut Dec 16 '14 at 16:52
  • @Jean-Claude Arbaut: I made a quick derivation of Taylor's theorem the other day to explain the theorem to a student in a simpler way. Of course, I knew that I would not get the same remainder. I didn't know it was called Schlömilch remainder though. Thanks for that. – Raskolnikov Dec 23 '14 at 13:03
  • This shows a novel application of the integral form of the remainder: https://math.stackexchange.com/questions/941222/evaluating-sums-and-integrals-using-taylors-theorem –  Feb 05 '18 at 22:54

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