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\begin{center}
\vskip 1cm{\LARGE\bf
Integer Sequences, Functions of Slow Increase\\
\vskip .1in
and the Bell Numbers}
\vskip 1cm
\large
Rafael Jakimczuk\\
Divisi\'on Matem\'atica \\
Universidad Nacional de Luj\'an\\
Buenos Aires \\
Argentina\\ 
\href{mailto:jakimczu@mail.unlu.edu.ar}{\tt jakimczu@mail.unlu.edu.ar}\\
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\centerline{In memory of my sister Fedra Marina Jakimczuk (1970--2010)} 

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\begin{abstract}
In this article we first prove a general theorem on integer sequences
$A_n$ such that the following asymptotic formula holds,
\[\frac{A_{n}}{A_{n-1}}\sim C n^{\alpha} f(n)^{\beta},\] where $f(x)$
is a function of slow increase, $C>0$, $\alpha>0$ and $\beta$ is a real
number.

We also obtain some results on the Bell numbers $B_n$ using well-known formulae. 
We compare the Bell numbers with $a^n$ $(a>0)$ and $(n!)^h$ $(0<h\leq 1)$. 

Finally, applying the general statements proved in the article we
obtain the  formula \[B_{n+1}\sim
e\ \left(B_n\right)^{1+\frac{1}{n}}.\]
\end{abstract}

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