Equivalence statement of Ratio Test 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.

Take $a_n=1$ to see that the converse is not true. 
i) means that the sequence $\{|a_n|\}$ is increasing, but it has to converge to $0$ in order to have convergence of the series $\sum_{n=1}^{+\infty}a_n$, so $a_n=0$ for $n$ large enough. But we had to assume that $a_n>0$ in order to i) make sense.
i) $\Rightarrow$ ii) is not true: take $a_n=2^{(-1)^n}$ for example.
