Re: [math-fun] Integers, rationals, algebraics, ?
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
The number of power series is the number of sequences of integers, which is uncountable. Andy On Feb 10, 2018 19:19, "Henry Baker" <hbaker1@pipeline.com> 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? —Dan
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participants (2)
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Andy Latto -
Henry Baker