Suppose $M$ is a square matrix with spectral radius <1. What can be said about $\|M^k\|$? Specifically, I hope to use $I+M+\cdots+M^k$ to approximate $(I-M)^{-1}$, and I hope to say something about approximation error (in proportional terms).
Edit: I am interested in "what can be said about $\|M^k\|$, and the size of approximation error (by truncating terms after $k$-th power), in terms of the spectral radius and $k$?"
If the norm in question is submultiplicative, then $\|M^r\|\le\|M\|^r$ for each nonnegative integer $r$ and hence $$ \left\|\sum_{r=0}^kM^r-(I-M)^{-1}\right\| =\left\|\sum_{r=k+1}^\infty M^r\right\| \le\sum_{r=k+1}^\infty\|M\|^r =\frac{\|M\|^{k+1}}{1-\|M\|}. $$