The Second Theorem of Consistency for Absolutely Summable Series

The Second Theorem of Consistency for summable series has been proved by Hardy in the following form: If

(i) the series Σc_n is summable (λ, k) to the sum C,

(ii) μ is a logarithmico-exponential function of λ such that μ = O(λ^ρ),

where Δ is a constant, then, the series Σc_n is summable (μ, k) to the sum C.

The object of this paper is to prove the corresponding theorem for absolutely summable series.