The first thing you have to realize about proving Taylor's theorem is that there are infinitely many versions of Taylor's theorem: one for each possible expression of the remainder term. In other words, what a Taylor's theorem really is is a proof that a certain expression involving n gives the nth remainder term, i.e. the diffierence between the nth Taylor polynomial and the function it approximates.
With that in mind we can motivate the Lagrange remainder form by (A) setting out to simply find some expression for the remainder, (B) generalizing our first attempt and discovering an infinite class of expressions for the remainder, and finally (C)
settling on the Lagrange form since it is somehow the most beautiful and compelling form in this class (or, on more pragmatic grounds, since it promises to be the most useful).
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