of DOR has been obtained [@JF:09].

Kunz and Shapiro’s work has established a universal $1/\Gamma$ law for arbitrary strength of disorder in a semi-infinite chain. Numerically one can verify $1/\Gamma$ law of the DOR, similar to what has been done by Terraneo and Guarnery [@GG:00] in finite samples for $P(\Gamma)$. Such verifications require the localization lengths to be much smaller than the size of the sample. In case of weak disorder an analytic result for the localization lengths is particularly useful. It comes from a second order perturbation theory. It states that the localization length is maximum near the middle of the energy band and is proportional to $W^{-2}$ where $W$ is the width of the disorder [@thou; @krammac; @KWegner:1981]. On the other hand, this result also leads to an interesting limiting situation where the localization lengths are much longer than the sample size. This is what we refer to as a weak disorder limit in this paper. This limit has scarcely been studied hitherto although it is relevant in the study of localization through resonances. Besides, there has been a believe for some sort of universality in the weak disorder limit. In this paper we address to this limit 