We establish conditions for the asymptotic relation $ln μ(x,F) ∼ ln μ(x,F_w)$ as $x → +∞$  utside of some exceptional set of finite Lebesgue measure for a positive functional series of the form $F(x) = \sum_{n=0}^{+∞} a_ne^{xλ_n + \tau(x)β_n}, a_n ≥ 0, (n ≥ 0),$ convergent for $x ≥ 0,$ where $\tau(x)$  is a positive increase differentiable function such that $\tau'(x) ≥ 1  (x > 0),(λ_n), (β_n)$ are positive sequences, $F_w(x) = \sum_{n=0}^{+∞} a_ne^{w(λ_n + β_n) +  xλ_n + \tau(x)β_n}, μ(x,F) = max\{a_ne^{xλ_n + \tau(x)β_n} : n ≥ 0\}$, and $w(t)$  is an increasing to $+∞$ in interval $[0,+∞)$ function.

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