  Let $G$ be a graph on $n$ vertices and $\lambda_1\geq \lambda_2\geq \ldots
\geq \lambda_n$ its eigenvalues. The Estrada index of $G$ is defined as
$EE(G)=\sum_{i=1}^n e^{\lambda_i}.$ In this work, we using an increasing
sequence converging to the $\lambda_1$ to obtain an increasing sequence of
lower bounds for $EE(G)$. In addition, we generalize this succession for the
Estrada index of an arbitrary nonnegative Hermitian matrix.
