Well, maybe not. Diagonalize the power series... At 04:17 PM 2/10/2018, Henry Baker wrote:
Counting argument?
C is uncountable; power series are countable?
At 02:45 PM 2/10/2018, Dan Asimov wrote:
Yes, I mean the coefficients of a power series about 0. I don't understand the comment about diverging at 1. But fair enough, asking for a specific question. Specific question: ------------------ Suppose there is an infinite power series f(z) = Sum_{0<=n<oo} c_n z^n that has integer coefficients {c_n in Z| n >= 0} with infinitely many c_n nonzero, and that converges for |z| < R, some R > 0. Let r be a non-algebraic zero of analytic function f(z) in the interior of its region of convergence: ----- f(r) = 0, |r| < R but P(r) != 0 for all nontrivial integer polynomials P(z). ----- SPECIFIC QUESTION: Does there exist a non-algebraic number in C that is *not* such a zero of an infinite power series with integer coefficients? ÂDan