Since 
$$
\frac{1}{1-q^{i_{1}+i_{2}}}\frac{1}{1-q^{i_{2}+i_{3}}}\ldots\frac{1}{1-q^{i_{n}+i_{1}}}>1,
$$
we find
\begin{align*}
f_{n}(q) & >\sum_{i_{1}\ldots i_{n}\geq1}q^{i_1+\ldots+i_n}\\
 & =\left(\sum_{i\geq1}q^{i}\right)^{n}\\
 & =\left(\frac{1}{1-q}\right)^{n}\to\infty.
\end{align*}