Proposition 4 ("logarithmic criterion of order 0")

Tag 08G2, permanent. Cite it as [Bourbaki, Tag 08G2].

Functions of a Real Variable, chapter V, § 4. APPLICATION TO SERIES WITH POSITIVE TERMS

Let $(u_n)$ be a series with positive terms; if $u_n \preccurlyeq n^\mu$ for some $\mu < -1$ then the series $(u_n)$ converges; if $u_n \succcurlyeq n^\mu$ for some $\mu \geq -1$ 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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