Remark 1

As a particular instance of Raabe's test one sees that if $\limsup_{n \to \infty} u_{n+1}/u_n < 1$ the series $(u_n)$ converges; if, on the other hand, $\liminf_{n \to \infty} u_{n+1}/u_n > 1$, the series $(u_n)$ has an infinite sum (d'Alembert's test).