When is a C∞ function analytic?

Students often wonder why their teachers insist on proving that the remainder in a Taylor expansion approaches zero before they will accept that the series represents the function from which it was obtained. The students tend not to be enlightened by the usual practice of invoking the Lagrange form of the remainder, with its mysterious unspecified intermediate point. It is not only students who are confused: a quite distinguished mathematician (A. Pringsheim) once made a mistake about remainders. This article is the story of the legacy of that mistake, which remained unnoticed for forty years. The Taylor series of a C" function f can have two kinds of singular behavior: the series may diverge except at its center, or it may converge, in a neighborhood of its center, to a function that differs from f in arbitrarily small neighborhoods of the center. There are many examples of the first kind of singularity, although they are not easy to show to an elementary class. For the second kind, the standard example is the function F defined by F(x) = exp( 1/x 2) for x ~ > 0 for all x in an interval. Is such a function real-analytic (equal to its Taylor series), or might it have a singularity of the second kind? This question was apparently first asked by Pringsheim [6] in 1893. He presented a proof that such a function f is necessarily analytic; I quote it in somewhat modernized notation. Suppose that