We present a method for estimating the asymptotic behavior of: $$ f^\alpha(x):=\sum_{n=1}^\infty n^\alpha l_n a_n x^n, x\to \infty, \alpha \in R, $$ related to a given entire function $f(x):=\sum_{n=1}^\infty a_n x^n$ of finite order $\rho$, $0<\rho<+\infty$, $a_n\ge 0$, $n\in N$; where $(l_n)$, $n\in N$, are slowly varying sequences in Karamata's sense.