Thanks, Gene. Apologies to Andy whose comment I had not understood. OK — let's try the same question with rational coefficients: Let (*) f(z) = Sum_{0 <= 0 < oo} c_n z^n where each c_n is in Q and infinitely many c_n are nonzero. If for some non-algebraic number r in C and some f(z) like (*) satisfy f(r) = 0 with r in the interior of the region of convergence of f(z), we say r is a *non-algebraic root* of f(z). New Question: ------------- Does there exist any non-algebraic number that is *not* a non-algebraic root? —Dan Gene Salamin wrote: ----- A power series with integer coefficients c[n], infinitely many of which are non-zero, has a radius of convergence at most 1. This is so because if c[n] ≠ 0, then for |z| ≥ 1, |c[n] z^n| ≥ 1 On Saturday, February 10, 2018, 2:46:42 PM PST, Dan Asimov <dasimov@earthlink.net> wrote: Yes, I mean the coefficients of a power series about 0. I don't understand the comment about diverging at 1. ------