With a view to better understanding real Taylor series, I have examined some books on basic Calculus, with an eye for the proofs of the Taylor series theorem and the possible authors' comments on its derivation. (My reaction when I first saw a proof of it, many years ago, was a mixture of great surprise and anxiety. And still, while I understand the individual steps, the way they all combine to produce e.g. the series for sinx strikes me as little short of miraculous.)
Up to now, from the books I have seen, I get the same impression: that this theorem is a technical exercise in repeated applications of the mean-value theorem. And we are lucky that some useful functions happen to have all derivatives bounded, so the remainder tends to zero and a nice series occurs, with nothing else to be said. But some authors do place some comments close to what I feel, albeit not very encouraging, e.g:
from Calculus, by Karl Menger: "Taylor's formula (...) is one of the great marvels of mathematics. (...) This is something like a mathematical action at a distance (...)"
from Real Analysis, by Laczkovich & Sós: "The statement of Theorem (...) is actually quite surprising (...) the derivatives of f at a alone determine the values of the function at every other point (...)"
from Introduction to the Calculus, by Osgood: "(...) Since it took the race two centuries to develop this formula after the Calculus was invented, the student will not be surprised that the reasons which underlie it cannot be given him in a few words. Let him accept it as a deus ex machina."
Now all this inquiry may be overly romantic and obsessive on my part, and Taylor series be a perfect example of the
"cold and austere beauty of mathematics" as Russell has expressed. But I think that sharing mental experiences helps
the mind to improve its turns and horizons, so may I ask:
What was your reaction when you first saw this theorem? And has your general understanding of it changed ever since, by some other way of looking at it and proving it?