Convergence Rates of Subseries

2018/05/25 by Leonetti, Paolo
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary: 40A35. Secondary: 26A12

paper · doi:10.48550/arxiv.1805.10366

Abstract

Let (x_n) be a positive real sequence decreasing to 0 such that the series ∑_n x_n is divergent and \liminf_n x_n+1/x_n>1/2. We show that there exists a constant θ∈ (0,1) such that, for each ℓ>0, there is a subsequence (x_nk) for which ∑_k x_nk=ℓ and x_nk=O(θ^k).

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