Convergence Rates of Subseries. Issue 2 (7th February 2019) - Leonetti, Paolo
Abstract: Let( x n ) be a positive real sequence decreasing to 0 such that the series∑ n x n is divergent andlim inf n x n + 1 / x n > 1 / 2 . We show that there exists a constantθ ∈ ( 0, 1 ) such that, for eachℓ > 0, there is a subsequence( x n k ) for which∑ k x n k = ℓ andx n k = O ( θ k ) .