My textbook and Wikipedia similarly state:
Say
$r=\liminf\limits_{n\to\infty}\,\left\lvert\dfrac{u_{n+1}}{u_n}\right\rvert\quad$ and $\quad R=\limsup\limits_{n\to\infty}\,\left\lvert\dfrac{u_{n+1}}{u_n}\right\rvert\;$.
If $R < 1$, the series converges,
If $r > 1$, the series diverges,
If $r\leqslant1\leqslant R$, the test is inconclusive, except if for a sufficiently large $n: \left\lvert\dfrac{u_{n+1}}{u_n}\right\rvert\geqslant1$, when the series diverges.
How can the second bullet be false, and the exception be true at the same time? (i.e. how is the exception not redundant?)
