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As this is special-functions question, I feel justified in using a bit of heavy artillery. Here goes nothing... In effect, what the OP seems to want to do is to evaluate $$\sum_{n=1}^\infty \frac{(q^{n+1};q)_\infty}{(q^n;q)_\infty} q^{n-1}$$ where $(a;q)_n$ is the $q$-Pochhammer symbol by approximating it with its partial sums. However, there is a more ...

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