Sums of Integer Powers via the Stolz-Cesàro Theorem

Some history. The treatment of the natural logarithm cases S(e, k, p) by means of the Integral Test is very well known. For example, they are treated in the classic encyclopedic work of Konrad Knopp, where 4 distinct proofs are given, one of which uses Cauchy’s Condensation Test. That particular proof can easily be modified to do the cases S(b, k, p) when b ≥ 2 but not the cases when 1 < b < 2 [3].

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