For any groupoid, it's groupoid cardinality is the sum of the reciprocals of the automorphism groups over the isomorphism classes. Let us consider the category of vector spaces over a finite field $\mathbb F_q$ with only invertible morphisms allowed.

Then, the size of the automorphism groups are $g_n= \prod_{i=1}^n(q^n-q^{i-1}) = q^{n\choose 2}\prod_{i=1}^n(q^i-1)$ so it's groupoid cardinality is the following infinite sum: $$\sum_{n \geq 1}\frac{1}{g_n}.$$

Is there a closed form expression for it? Note that as $q\to 1$ in an appropriate sense, the general linear group is supposed to converge to the symmetric group and the groupoid of vector spaces should converge to the groupoid of finite sets.

In this sense, the above infinite sum is an analog of the usual one for $e$ and can perhaps reasonably be called a q-analog of $e$.

Alternatively, in the case of *abelian*, (even semi-simple) categories, is there a better replacement for the groupoid cardinality that I should be using?