How are poles and orders of poles of formal power series defined?
The particular case, I am interested in, is the following definition from [Atiyah-Macdonald, Introduction to commutative algebra, p.116]:
Let $A$ be a graded Ring, $M$ a finitely generated graded $A$-module and $\lambda$ an additive function on the class of finitely generated $A_0$-modules (e.g. the length-function). Then the Hilbert-Poincaré series $P(M,t) \in \mathbb{Z}[[t]]$ is defined as $$ P(M,t) := \sum_{n=0}^\infty \lambda(M_n) t^n. $$ Now $d(M)$ is defined to be the order of the pole of $P(M,t)$ at $t=1$. What exactly is $d(M)$?
Thank you in advance!
By Theorem 11.1, we know that $\rm\:P(M,t)\:$ is a rational function. Recall their notion of pole: if $\rm\: P(t) = f(t)/((t-1)^n g(t)),\:$ with $\rm\: f(1),g(1)\ne 0\:$ then $\rm\,P\,$ has a pole of order $\rm\:n\:$ at $\rm\:t = 1.\:$