Proposition 5 ("logarithmic criterion of order $p$")

Let $(u_n)$ be a series with positive terms. If $u_n \preccurlyeq \frac{1}{nl_1(n)l_2(n)\ldots l_{p-1}(n)(l_p(n))^\mu}$ for some $\mu > 1$ then the series $(u_n)$ converges; if $u_n \succcurlyeq \frac{1}{nl_1(n)l_2(n)\ldots l_{p-1}(n)(l_p(n))^\mu}$ for some $\mu \leq 1$ then the series $(u_n)$ has an infinite sum.

If $0 \leq q < 1$ one has $q^n \preccurlyeq n^{-\mu}$ for any $\mu > 0$; applying the logarithmic criterion of order 0 again proves the convergence of the geometric series $\sum_{n=0}^\infty q^n$ for $|q| < 1$ (Gen. Top., IV, p. 364). If one applies prop. 1 with $v_n = q^n$ one obtains a criterion which may be put in the following form ("Cauchy's criterion"): Let $(u_n)$ be a series with positive terms; if $\limsup_{n \to \infty}(u_n)^{1/n} < 1$ then the series $(u_n)$ converges; if $\limsup_{n \to \infty}(u_n)^{1/n} > 1$ then the series $(u_n)$ has infinite sum. Indeed, if $\limsup_{n \to \infty}(u_n)^{1/n} = a < 1$ then $u_n \preccurlyeq q^n$ for every $q$ such that $a < q < 1$. If, on the other hand, $\limsup_{n \to \infty}(u_n)^{1/n} = a > 1$ then $u_n \succcurlyeq q^n > 1$ for infinitely many values of $n$, for any $q$ such that $1 < q < a$; since $u_n$ does not tend to 0 one has $\sum_{n=1}^\infty u_n = +\infty$.

This criterion is very useful in the theory of entire series, which we shall study later; but it even so it does not permit one to decide the convergence of the series $(1/n^\alpha)$, in other words, its field of application is much more restricted than that of the logarithmic criteria.