= Solution
{c}
For an <asymptotic sequence> $a_{n+1}(\epsilon)=o(a_n(\epsilon))$, the <asymptotic expansion> condition is that every fixed truncation has remainder smaller than its last retained term:
$$
\boxed{f(\epsilon)-\sum_{n=0}^Na_n(\epsilon)=o(a_N(\epsilon))\quad\text{for every fixed }N.}
$$
This limit is taken as $\epsilon\to0$ in the specified direction or sector. The number $N$ stays fixed while taking that limit. The infinite series need not converge for any nonzero parameter.
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