Bourbaki

Exercise 5, § 4

Functions of a Real Variable, V, p. 50

Let (un)(u_n) be a series with terms >0> 0 from a certain rank on. Prove that if

lim sup⁡n→∞(un+1un)n<1e, \limsup_{n \to \infty} \left( \frac{u_{n+1}}{u_n} \right)^n < \frac{1}{e},
the series (un)(u_n) is convergent, and that if lim inf⁡n→∞(un+1/un)n≥1/e\liminf_{n \to \infty} (u_{n+1}/u_n)^n \geq 1/e, the series has an infinite sum.

Solution

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Machine translation of the English, by gpt-5-mini. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.