Proposition 8

Tag 08G9, permanent. Cite it as [Bourbaki, Tag 08G9].

Let $(u_n)$ be a series with terms $> 0$ from a certain point on. If, from a certain point, one has

\frac{u_{n+1}}{u_n} \leq 1 - \frac{1}{n} - \frac{1}{nl_1(n)} - \ldots - \frac{1}{nl_1(n)l_2(n)\ldots l_{p-1}(n)} - \frac{\alpha}{nl_1(n)l_2(n)\ldots l_p(n)}

for some $\alpha > 1$ , then the series $(u_n)$ converges; if, from some point on, one has

\frac{u_{n+1}}{u_n} \geq 1 - \frac{1}{n} - \frac{1}{nl_1(n)} - \ldots - \frac{1}{nl_1(n)l_2(n)\ldots l_p(n)}

then the series $(u_n)$ has an infinite sum.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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