Indeed, I conjecture that g(z) (as defined below) and all its derivatives converge at every point of the unit circle in the complex plane; then writing z = exp(i t) gives a periodic function of real variable t, whose real part h(t) = Re(g(exp(i t))) is given by a Fourier series which will no doubt be Cesaro summable. This function might be a simpler example of the type I mentioned earlier; but it may well have discontinuities, as do the standard Fourier series examples --- square wave, sawtooth, etc. Can somebody whose analysis is less rusty than mine cast some light here? WFL On 9/11/06, 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.
Good point, Richard, and the limit as x -> 0 is not that interesting, either.
I posted the wrong question, yet another mistake created by undue haste.
*****The question of interest is: "What happens to g(x) as x -> 1- ?"
--Dan