(Oops, sorry, Gene, you're right--I completely missed the 1/n!. Unbelievable that something with that smooth an edge could have a curtain of poles there. Even less believable with the edge hidden by your challenge continuation. Nice plot. only one nontrivial root.)
Foo, there are no poles. You can write in closed form the Taylor expansion coefficients for a point on the circle. The only problem is the radius of convergence is 0. So natural boundaries have nothing to do with curtains of poles--they're just places you can't continue past because of vanishing convergence circles. I had no idea of the inequivalence. But then why do the radii vanish? Does this mean there are infinitely many solutions to Gene's challenge? Looks like there's also a nice Fourier series on the circle. --rwg