# 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/$\sim$