Abstract
In this paper we will extend the high indices Theorem by Hardy-Littlewood, which says that an Abel summable series, where the exponents fulfil a Hadamard gap criterion, is convergent and it converges to its Abel sum. The focus here is concerning a class of summability methods and their critical rate of convergence, i.e. for a given summability method what is its rate of convergence implying that the series must be identically constant.
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