From: "Achava Nakhash, the Loving Snake" Subject: Re: Fiddley Maclaurin series convergence question Date: Fri, 22 Jan 1999 10:42:04 -0800 Newsgroups: sci.math To: eric.fennessey@gecm.com Keywords: Taylor series at endpoints, Abel summation as Stieltjes integrals eric.fennessey@gecm.com wrote: > Hi, > > I have a question I hope some clever analyst can answer, proof or > counterexample would be fine. > > Suppose f:[a,b]->IR is continuous and has a Maclaurin series expansion on > (a,b). Suppose the Maclaurin series converges at b, does it follow that it > converges to f(b). > > The question arose from a piece of work in which I had a Maclaurin series > expansion for a function f on (-0.25,0.25). The series converged at 0.25 (the > radius of convergence) and I wanted to know whether it would necessarily > converge to f(0.25). > > In the particular instance I showed this was actually true, and was wondering > if it was true in general, or if there was a nifty counter-example. > > Thanks in advance, > Eric I am not an analyst of any description, but here is what I remember about this very interesting question. If a power series converges on an interval, say (a - r, a + r), just a reminder that convergence is symmetrical around the number a when the series is in terms of powers of (x - a), and if the power series also converges at one of the endpoints, then indeed the function is continous as you approach the endpoint from insided the interval. In other words your conjecture is correrct. This is a surprsingly difficult fact to prove. The proof uses a beautiful trick known as Abel's partial summation, so presumably the theorem is due to Abel. It should be easy to find references to Abel's partial summation in any good introductory analysis book such as that by Rudin that was standard back when I was a student. For those familiar with Stieltjes integration (also covered in Rudin) the Abel partial summation is a highly useful example of integration by parts with the right Stieltjes integration. That having been said, the direct proof of Abel's partial summation could be followed by a motivated high school student. This is one of those things in mathematics that is easy to see once someone has shown it to you, but it took a genius to see not only the idea but also its usefulness. Hope this helps, Achava