MathDB
Miklos Schweitzer 1970_10

Source:

October 22, 2008
limitreal analysisreal analysis unsolved

Problem Statement

Prove that for every ϑ \vartheta, 0<ϑ<1 0<\vartheta<1, there exist a sequence λn \lambda_n of positive integers and a series ∑n=1∞an \sum_{n=1}^{\infty} a_n such that (i)λn+1−λn>(λn)ϑ \lambda_{n+1}-\lambda_n > (\lambda_n)^{\vartheta}, (ii) lim⁡r→1−∑n=1∞anrλn \lim_{r\rightarrow 1^-} \sum_{n=1}^{\infty} a_nr^{\lambda_n} exists, (iii) ∑n=1∞an \sum _{n=1}^{\infty} a_n is divergent. P. Turan
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