In his Paper The Euler Characteristic of acyclic Categories Kazunori Noguchi gives the following definition:
Let $f(t)$ be a formal power series over $\mathbb{Z}$. If there exists a rational function $g(t)/h(t)$, such that $f(t)=g(t)/h(t)$, then define $f_{|_{t=-1}} = g(-1)/h(-1)$ if $h(-1)\not= 0$
Now - given my pretty limited knowledge about formal power series - I'd assume that $f(t)=g(t)/h(t)$ yields $f_{|_{t=-1}} = g(-1)/h(-1) = f(-1)$ which makes me wonder why he's using the quotient construction in the first place, instead of just evaluating $f(t)$ at $t=-1$ (provided f(t) converges at $t=-1$).