On radii of convergence of power series. (English) Zbl 0886.40002

Let \(s\) be the set of all real sequences \(a=(a_n)^\infty_0\) and let \(\sum^\infty_0 a_nx^n\) be a power series. Let \(r,\sigma: s\to[0,\infty]\) be two functions defined by the rules \[ r(a)=1/ \Bigl(\lim_{n\to\infty} \sup \root n\of {|a_n|} \Bigr) \quad \text{and} \quad \sigma(a) =\lim_{n\to\infty} \sup \root n\of {|a_n|}. \] Writing this paper the authors were motivated by the papers P. Kostyrko and T. Šalát [Rend. Circ. Mat. Palermo, II. Ser. 31, 187-194 (1982; Zbl 0502.40001)] and T. Šalát [Czech. Math. J. 34(109), 362-370 (1984; Zbl 0558.40001)] in which exponents of convergence of nondecreasing sequences were investigated. In the present paper the authors study the fundamental properties of the functions \(r\) and \(\sigma\) in detail and investigate the radii of convergence of subseries of a given series. Some of the theorems proved are: (i) Each of the functions \(\sigma,r\) is surjective. (ii) Each of the functions \(\sigma,r\) is discontinuous everywhere and belongs exactly to the second Baire class. (iii) Almost all subseries \(\sum^\infty_{n=0} \varepsilon_na_nx^n\) of a given power series \(\sum^\infty_{n=0} a_nx^n\) have the same radius of convergence as the series \(\sum^\infty_{n=0} a_nx^n\), where \((\varepsilon_n)^\infty_0\) is a sequence of 0’s and 1’s with infinitely many \(\varepsilon_n\) equal to 1. Finally they study analyticity of infinitely many differentiable functions and obtain structures of certain sets.


40A30 Convergence and divergence of series and sequences of functions
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)