= Solution
Iterate the correction: after constructing $b_{j-1}\in S^{-(j-1)}$, set
$$
a_j=a\,b_{j-1}\in S^{-M-(j-1)}.
$$
The same calculation improves the remainder by one order:
$$
Q(x,\lambda+D_x)\sum_{j=1}^Na_j=1-b_N,\qquad b_N\in S^{-N}.
$$
Let $A_j$ and $B_N$ be the corresponding inverse <oscillatory integral>[oscillatory integrals]. Then
$$
\boxed{Q(x,D)(A_1+\cdots+A_N)=\delta_0-B_N}.
$$
When $N>n$, the frequency integral defining $B_N$ converges absolutely and depends continuously on $x$, so $B_N\in C(X)$. This completes the finite-order <parametrix> construction.
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