tail equivalence in the category of series

Consider the category Series.

Say that two objects of Series are tail-equivalent if and only if their quotient with respect to their variables (as the index is increased without bound) is one.

For instance \sum_{n=1}^{\infty} q^{n^{2}} and  \sum_{n=1}^{\infty} q^{n^{2}-q^{n}} 

We’ll write \sum_{n=1}^{\infty} q^{n^{2}} \sim \sum_{n=1}^{\infty} q^{n^{2}-q^{n}}

I believe \sim is an equivalence relation. But lo, what of the quotient category Series/\sim


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