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In this paper we shall consider the growth at infinity of a sequence $(P_n)$of entire functions of bounded orders. Our results extend the results in\cite{trong-tuyen2} for the growth of entire functions of genus zero. Given asequence of entire functions of bounded orders $P_n(z)$, we found a nearlyoptimal condition, given in terms of zeros of $P_n$, for which $(k_n)$ that wehave \begin{eqnarray*} \limsup_{n\to\infty}|P_n(z)|^{1/k_n}\leq 1\end{eqnarray*} for all $z\in \mathbb C$ (see Theorem \ref{theo5}). Exploringthe growth of a sequence of entire functions of bounded orders lead naturallyto an extremal function which is similar to the Siciak's extremal function (SeeSection 6).

The growth at infinity of a sequence of entire functions of bounded orders