For differentiable functions, one would think that the values of successive derivatives at a single point are so “very local” properties of a function, that they cannot possibly determine its values at other points lying “very far away”. As is well-known, for many useful functions this is not so, as revealed by the Taylor-series expansion theorem, of functions exponential, trigonometric, logarithmic etc.
My problem is that in all proofs I know of, this overwhelmingly wonderful & miraculous phenomenon is seemingly extinguished by a rather dry technical manoeuver: by repeated application of the mean-value theorem, for all x the remainder is bounded by something tending to zero as n tends to infinity, so the expansion is valid.
I wonder, where has the “miracle-point” gone? Can someone give an intuitive explanation that captures the essence of these “by-a-single-point-uniquely -determinable” functions?