The author studies the formal power series (everywhere divergent) \(F(z)=\sum^{\infty}_{r=1}a_ rz^ r\), where \(a_ r=r^ pw(r)(r!)^ m,\) \(p\geq 0\), \(m\geq 1\) are integers and w(r) is such that for some \(\sigma >0\), \(w(r)\sim \sum^{\infty}_{i=0}w_ ir^{-i- \sigma},\) as \(r\to \infty\). It is shown that the partial sums \(A_ r(z)\) of the above series can be represented as \(A_{r-1}(z)=A(z)a_ rz^ rg(r,z)\) where A(z) is the Borel-type sum of F(z) (defined by the author as a generalization of the Borel sum of F(z)) and for the converging factor g(r,z) an asymptotic expansion for \(r\to \infty\) and fixed z is obtained.