  We consider continued fractions
\frac{-a_1}{1-\frac{a_2}{1-\frac{a_3}{1-...}}} \label{fr} with real
coefficients $a_i$ converging to a limit $a$. S.Ramanujan had stated the
theorem (see [ABJL], p.38) saying that if $a\neq\frac14$, then the fraction
converges if and only if $a<\frac14$. The statement of convergence was proved
in [V] for complex $a_i$ converging to $a\in\mathbb
C\setminus[\frac14,+\infty)$ (see also [P]). J.Gill [G] proved the divergence
of (\ref{fr}) under the assumption that $a_i\to a>\frac14$ fast enough, more
precisely, whenever \sum_i|a_i-a|<\infty.\label{gill} The Ramanujan conjecture
saying that (\ref{fr}) diverges always whenever $a_i\to a>\frac14$ remained up
to now an open question. In the present paper we disprove it. We show (Theorem
\ref{th1}) that for any $a>\frac14$ there exists a real sequence $a_i\to a$
such that (\ref{fr}) converges. Moreover, we show (Theorem \ref{go}) that
Gill's sufficient divergence condition (\ref{gill}) is the optimal condition on
the speed of convergence of the $a_i$'s.

Dynamical Systems;Complex Variables