Exercise 5, § 4

Let $(u_n)$ be a series with terms $> 0$ from some point on. Show that if

then the series converges, and if $\liminf_{n \to \infty} \left( \frac{u_{n+1}}{u_n} \right)^n \geqslant \frac{1}{e}$ the series has an infinite sum.