Proposition 5 ("logarithmic criterion of order 0")

If on a neighbourhood of $\alpha$ one has $f(x) \leq 1/(x-\alpha)^{\mu}$ for some $\mu < 1$ then the integral $\int_{\alpha}^{a} f(t) \, dt$ converges; if $f(x) \geq 1/(x-\alpha)^{\mu}$ for some $\mu \geq 1$ then the integral $\int_{\alpha}^{a} f(t) \, dt$ is infinite.

We leave it to the reader to translate the logarithmic criterion of order $n$ similarly.

The application of the logarithmic criteria is immediate if one knows how to obtain the principal part of $f$ with respect to a comparison scale containing the functions that feature in these criteria: if $f_1$ is the principal part, the integral $\int_{\alpha}^{+\infty} f(t) \, dt$ converges or is infinite together with $\int_{\alpha}^{+\infty} f_1(t) \, dt$, and the logarithmic criteria apply immediately to this last integral.