Radii of convergence of power series
The power series \(\sum _{n=0}^\infty a_n x^n\) is represented by the sequence \((a_n)_{n=0}^\infty \) in the Fréchet metric space of sequences with the metric \[ d ( (a_n)_{n=0}^\infty , (b_n)_{n=0}^\infty) = \sum _{n=0}^\infty 2^{-n} \frac{|a_n - b_n|} {(1+|a_n-b_n|)}. \] \(\sigma \)-porous properties of the set of sequences with positive radius of convergence are proved. Moreover, the authors study the Hausdorff dimension of the set of infinite subseries with different radius of convergence than a given series. (Subseries are identified with the numbers from \((0,1]\) such that the \(n\)th summand is contained in the subseries iff 1 is at the \(n\)th position of the dyadic expansion).
scientific article; zbMATH DE number 1449516