1.Series $\sum a_n$ diverges if (i) there exists $N\in \mathbb{N}$ such that $n≧N ⇒ |a_{n+1}/a_n|≧1$
2.Series $\sum a_n$ diverges if (ii)$\limsup |a_{n+1}/a_n|>1$
Here, both 1&2 are true. It is easy to see that (ii) implies (i), but is the converse true? Is 2 more general than 1?
Plus, it's usual that root test is harder to apply than ratio test, but i think root test is easier when a sequence $\{a_n\}$ has infinite 0 terms, since $|a_{n+1}/a_n|$ cannot be defined. Am i right or is there a "trick" to avoid this?
EDIT: I just noticed that this post is wrong. Statement 2 is indeed false.