Proposition 4 ("logarithmic criterion of order n")

Let f be a regulated function $\geq 0$ on an interval $[a, +\infty[$; if, for some $\mu > 1$, $f(x) \preccurlyeq \frac{1}{x l_1(x) l_2(x) \ldots l_{n-1}(x) (l_n(x))^\mu}$, then the integral $\int_a^{+\infty} f(t) \, dt$ converges; if $f(x) \succcurlyeq \frac{1}{x l_1(x) l_2(x) \ldots l_{n-1}(x) (l_n(x))^\mu}$ for some $\mu \leq 1$, then the integral $\int_a^{+\infty} f(t) \, dt$ is infinite.

Each logarithmic criterion is thus applicable to functions for which the criteria of lower order are not decisive (cf. V, p. 264, exerc. 5 b) and V, p. 265, exerc. 8).

Because of its usefulness we translate the criterion of order 0 for integrals $\int_{\alpha}^{a} f(t) \, dt$ where $f$ is regulated and $\geq 0$ on a non-compact interval $]\alpha, a]$: