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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 $t=-1$).

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    Ah, but as in GEdgar's answer, the series for $f$ need not converge to be evaluable. This is a kind of "algebraic" analytic continuation.2012-10-31

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Example. $f(x) = \sum_{n=0}^\infty x^n$, $g(x)=1$, $h(x) = 1-x$. Then $f(x) = g(x)/h(x)$, but $f(x)$ diverges at $-1$. Nevertheless, we want to plug in $-1$ in the rational function and get $ f_{|_{t=-1}} = g(-1)/h(-1)=1/(1+1)=1/2. $ But despite this, we do not write this calculation as $ f(-1) = 1 - 1 + 1 - 1 + 1 - 1 + \dots = \frac{1}{2} $

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    Ah ok, I see. Thank you.2012-11-01