A further note on differentials

~{f (x).x}^-kf(x) when x>X, (7) ax then, as a little calculation shows, f(x)^Ax-~, (8) where A is a positive constant. No one would prefer (7) to (8) as a criterion of convergence and (8), like (6), is a well-known test for the convergence of infinite integrals. The next test, in the usual order, is given by taking (x)=x log x in Theorems 1 and 2. That the test is useless may be seen from the fact (mildly interesting in its proof) that