  Recently, Ono and the third author discovered that the reciprocals of the
theta series $(q;q)_\infty^3$ and $(q^2;q^2)_\infty(q;q^2)_\infty^2$ have
infinitely many closed formulas in terms of MacMahon's quasimodular forms
$A_k(q)$ and $C_k(q)$. In this article, we use the well-known infinite product
identities due to Jacobi, Watson, and Hirschhorn to derive further such closed
formulas for reciprocals of other interesting infinite products. Moreover, with
these formulas, we approximate these reciprocals to arbitrary order simply
using MacMahon's functions and {\it MacMahon type} functions. For example, let
$\Theta_{6}(q):=\frac{1}{2}\sum_{n\in\mathbb{Z}} \chi_6(n) n
q^{\frac{n^2-1}{24}}$ be the theta function corresponding to the odd quadratic
character modulo $6$. Then for any positive integer $n$, we have
$$\frac{1}{\Theta_{6}(q)}= q^{-\frac{3n^2+n}{2}}\sum_{\substack{k=r_1\\ k\equiv
n\hspace{-0.2cm}\pmod{2}}}^{r_2}(-1)^{\frac{n-k}{2}}A_{k}(q)C_{\frac{3n-k}{2}}(q)+O(q^{n+1}),$$
where $r_1:=\lfloor\frac{3n-1-\sqrt{12n+13}}{3}\rfloor+1$ and
$r_2:=\lceil\frac{3n-1+\sqrt{12n+13}}{3}\rceil-1$.
