Artur Braun , the reason for my question is the limit superior form of the ratio test for real series.

I wonder if there exist actual cases where the limit superior form of the ratio test does not reduce to the simple form of the ratio test.

Actually, one can chose arbitrarily finitely many numbers in )0,1( to be contained in the set $abs(x_{n+1}/x_n)$. For this chose arbitrary $a_1, ..., a_{ m - 1} \in )0, 1($ and put for every $0 \le k < m$ and $p = 0, 1, ...., $

$x_{p m + k + 1} := \prod_{i=1}^k a_i^{p + 1} \prod_{i= k+1}^{m -1} a_i^p$. Here the required set contains all the a_i 's and all are less than 1.

I concur with Lahbib Oubbi