Postscript to "Convergence Theorems for Series of Positive Terms"

In the paper cited above, I have based the convergence theory of positive series on

If O = d₀ = D₀, O<d_n=D_n-D_n-1 (n = 1, 2,...), D_n→∞, and if F(x) is positive monotone decreasing, ∫ F(x)dx is convergent, then ∫d_nF(D_n) is convergent.

The first result of this note leads to the more general companion.