  For $x\in (0,1)$, let $\langle d_1(x),d_2(x),d_3(x),\cdots \rangle$ be the
Engel series expansion of $x$. Denote by $\lambda(x)$ the exponent of
convergence of the sequence $\{d_n(x)\}$, namely \begin{equation*} \lambda(x)=
\inf\left\{s \geq 0: \sum_{n \geq 1} d^{-s}_n(x)<\infty\right\}.
\end{equation*} It follows from Erd\H{o}s, R\'{e}nyi and Sz\"{u}sz (1958) that
$\lambda(x) =0$ for Lebesgue almost all $x\in (0,1)$. This paper is concerned
with the topological and fractal properties of the level set $\{x\in (0,1):
\lambda(x) =\alpha\}$ for $\alpha \in [0,\infty]$. For the topological
properties, it is proved that each level set is uncountable and dense in
$(0,1)$. Furthermore, the level set is of the first Baire category for
$\alpha\in [0,\infty)$ but residual for $\alpha =\infty$. For the fractal
properties, we prove that the Hausdorff dimension of the level set is as
follows: \[ \dim_{\rm H} \big\{x \in (0,1): \lambda(x) =\alpha\big\}=\dim_{\rm
H} \big\{x \in (0,1): \lambda(x) \geq\alpha\big\}= \left\{
  \begin{array}{ll}
  1-\alpha, & \hbox{$0\leq \alpha\leq1$;}
  0, & \hbox{$1<\alpha \leq \infty$.}
  \end{array} \right. \]
