  In this paper we show under weak assumptions that for
$R\stackrel{d}{=}1+M_1+M_1M_2+\ldots$, where $P(M\in[0,1])=1$ and $M_i$ are
independent copies of $M$, we have $\ln P(R>x)\sim C\, x\ln P(M>1-\frac1x)$ as
$x\to\infty$. The constant $C$ is given explicitly and its value depends on the
rate of convergence of $\ln P(M>1-\frac1x)$. Random variable $R$ satisfies the
stochastic equation $R\stackrel{d}{=}1+MR$ with $M$ and $R$ independent, thus
this result fits into the study of tails of iterated random equations, or more
specifically, of perpetuities.
