= Solution
{c}
For every fixed truncation index $N$, the remainder must be smaller than the last retained term:
$$
\boxed{f(\epsilon)-\sum_{n=0}^{N}a_n(\epsilon)
=o(a_N(\epsilon))\quad(\epsilon\to0).}
$$
This is an <asymptotic expansion> relative to the <sequence> in part (i). The number of retained terms is fixed before taking the limit. The associated <series> need not converge for any nonzero parameter; its role is to give successively smaller truncation errors. If $a_n$ is an <asymptotic sequence>, the same condition can be stated as a remainder $O(a_{N+1})$ when the next expansion condition is available, but the displayed <little o notation> formulation is the defining one.
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