On page 68 of Christopher Small's Expansions and Asymptotics for Statistics, we look at: $$ E(T_n)=\sqrt{\frac{8}{\pi n}}e^{-n/8}\Big(1-\frac{8}{n}+\frac{96}{n^2}-\cdots\Big). $$ (Here, $n$ is the sample size and $T_n$ is a function of the observed data.)
The authors then write:
In this example, the asymptotic rate of convergence of $T_n$ to zero is superexponential. However, the expansion is "$n$-asymptotic" in nature.
(The boldfacing is mine.)
Up to the page cited above, the boldfaced terms above are not defined rigorously. What do they refer to and could you please recommend a more elementary text that deal with (topics that eventually lead to) those terms?