I'm trying to derive the closed form of the fermionic coherent state defined by the relation:
$$ f_i|\vec{\eta}\rangle = \eta_i |\vec{\eta}\rangle \tag{4.10} $$
My book (Atland and Simons, Condensed Matter Field Theory, pages 160-163 suggests that I use the state
$$ |\vec{\eta}\rangle = \exp\left[ -\sum_i \eta_i f_i^{\dagger} \right]|0 \rangle\tag{4.17} $$
where the $\eta_i$ are Grassmann numbers: $\{\eta_i,\eta_j\}=0$. I've noticed that Atland and Simons regularly use the relation:
$$ \exp\left[ -\sum_i \eta_i f_i^{\dagger} \right] = \left(1-\sum_i \eta_i f_i^{\dagger}\right) $$
(see, e.g. footnote 3 on pg 163, or the solutions to problems on pg 181). Why is this equality correct? I would think that if we have, for example, $i$ ranging from $1$ to $2$ then:
$$ \exp\left[-\eta_1 f_1^{\dagger}-\eta_2f_2^{\dagger}\right] = 1 + \partial_{\eta_1}\exp\left[...\right]\eta_1 + \partial_{\eta_2}\exp\left[...\right]\eta_2 \\ + \frac{1}{2}\partial_{\eta_1}\partial_{\eta_2}\exp\left[...\right]\eta_2\eta_1 + \frac{1}{2}\partial_{\eta_2}\partial_{\eta_1}\exp\left[...\right]\eta_1\eta_2 $$
Noticing that the last two terms are the same by anticommutation relations we find that:
$$ \exp\left[-\eta_1 f_1^{\dagger}-\eta_2f_2^{\dagger}\right] = 1 - \eta_1 f_1^{\dagger} - \eta_2 f_2^{\dagger} + \eta_1\eta_2 f_1^{\dagger}f_2^{\dagger}$$
In general, the Taylor expansion of an exponential of a linear combination of $N$ Grassmann variables should have nonzero terms up to order $N$ in the Grassmann variables? Is there a reason that Altland and Simons completely ignore the terms of order $>1$?