Let f be a real-valued continuous function on \([0,\infty )\) . Although every convergent integral \(s(x)=\int _0^xf(t)dt\) is (H, 1) summable, the converse-deducing convergence from (H, 1) summability-requires Tauberian conditions. In this study, we demonstrate that the slowly decreasing condition ( \(\mathcal{S}\mathcal{D}\) ), which is more general than convergence, is sufficient to establish the convergence of integrals that are (H, 1) and (H, 2) summable. Our results extend the theorems in [1, 2] and introduce weaker Tauberian conditions applicable to (H, k) summability methods.
