On 9/10/06, Daniel Asimov <dasimov@earthlink.net> wrote:
Btw, on his website, Noam Elikies asks what the limit is of the infinite series
g(x) = x - x^2 + x^4 - x^8 + . . . +- x^(2^k) -+ . . .
(which converges for |x| < 1) as x -> 0-. It's a cool problem to let students try to guess the answer just using calculators.
--Dan
There was a series of slim books by Konrad Knopp entitled "Counterexamples in ..." --- the "Analysis" volume would no doubt give more relevant examples. The one quoted is an example of a function of a complex variable with a continuous frontier of singularities around the unit circle, so incapable of analytic continuation by standard infinite sum techniques --- I don't know whether continued fractions might be more successful? Recently, in the course of attempting to blend two functions along an interval while preserving all derivatives at the endpoints, I encountered a function with continuous derivatives of all orders on the real line, yet nowhere analytic there. [This horror may well constitute the frontier of some function analytic on the upper half plane, which I was unable to determine.] Anybody know of anything similar, but perhaps simpler? WFL