Rudin's Principles of Mathematical Analysis phrases the ratio test (Thm. 3.34) as follows:
"The series $\sum a_n$
converges if $\limsup_{n \to \infty} |a_{n+1}/a_n| < 1$,
diverges if $|a_{n+1}/a_n| \geq 1$ for $n \geq n_0$, where $n_0$ is some fixed integer.
If $$\liminf_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| \leq 1 \leq \limsup_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|,$$ the test gives no information."
My assumption was that for every series, exactly one of the conditions in (1)-(3) must hold.
From the phrasing this wasn't immediately obvious, and after thinking about it a bit I found that it isn't actually the case.
For example the series $1 + 1 + 1 + \cdots$ satisfies both (2) and (3).
I think we can fix this overlap by rephrasing the condition of (2) as $\liminf > 1$.
I like this better because it makes the conditions more symmetrical and it's obvious then that exactly one of the three must be satisfied in all cases.
Does this leave the theorem intact?
Edit: I think the change does leave the theorem true and fix the logical overlap, but weakens it because (2) is true as written and my replaced condition for (2) is narrower. Is there a way to keep the theorem true without weakening it but still fix the logical overlap in a simple way?