## 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.

### MSC:

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

### Citations:

Zbl 0502.40001; Zbl 0558.40001