Given a system of $n$ linear equations $$ x_i=\sum_{k=1}^{n}a_{ik}x_k+b_i \quad i=1,2,...,n$$ I'd like to employ the fixed point iteration method to find $x_i$. The fixed point iteration define $$ x_i^{N+1}=\sum_{k=1}^{n}a_{ik}x_k^{N}+b_i \quad i=1,2,...,n$$ with $x_i^{0}=0$. It states that, given that given the conditions of Banach fixed point system is satisfied, then $x_i^{\infty}=x_i$
What conditions are to be imposed upon $\sum_{k=1}^{n}a_{ik}$ to satsify Banach fixed point theorem? In particular I'm very interested in the case where $n=\infty$, that is a system of infinite equations.