  We derive asymptotic estimates for the coefficient of $z^{k}$ in $\left(
f\left( z\right) \right) ^{n}$ when $n\rightarrow \infty $ and $k$ is of order
$n^{\delta }$, where $0<\delta <1,$ and $f\left( z\right) $ is a power series
satisfying suitable positivity conditions and with $f\left( 0\right) \neq 0,$
$f^{\prime }\left( 0\right) =0.$ We also show that there is a positive number
$\varepsilon <1$ (easily computed from the pattern of non-zero coefficients of
$f\left( z\right) $) such that the same coefficient is positive for large $n$
and $\varepsilon <\delta <1$, and admits an asymptotic expansion in inverse
powers of $k$. We use the asymptotic estimates to prove that certain finite
sums of exponential and trigonometric functions are non-negative, and
illustrate the results with examples.
