We present a formula that turns power series into series of
functions. This formula serves two purposes: first, it helps to
evaluate some power series in a closed form; second, it transforms
certain power series into asymptotic series. For example, we find
the asymptotic expansions for λ>0 of the incomplete gamma function γ(λ,x) and of the Lerch transcendent Φ(x,s,λ). In one particular case, our formula reduces
to a series transformation formula which appears in the works of
Ramanujan and is related to the exponential (or Bell) polynomials.
Another particular case, based on the geometric series, gives rise
to a new class of polynomials called geometric polynomials.
