Confusion regarding summing a matrix series I have got following series while working out for an iteration problem.
$X_{k+1} = (1 + x + x^{2}+\cdots x^{2^{k+2}-3})$, $k = 0, 1, 2\cdots $ and $\|x\|<1$. My question is could I write $X_{k+1} = (1- x)^{-1}$, when $k \to \infty$. 
Thanks for the help and time.

Since $\|X\|<1$, you have $I+X+X^2+\cdots+X^n = (I-X)^{-1}(I-X^{n+1})$ (this is easy to verify noting that $I-X$ is invertible since all eigenvalues $\lambda$  of $X$ satisfy $|\lambda|<1$, and by multiplying both sides by $I-X$).
Since $\|X^n\| \le \|X\|^n$, it follows that $\lim_n X^n = 0$, from which it follows that $\lim_n \sum_{k=0}^{2^{k+2}-3} X^k =  \lim_n \sum_{k=0}^n X^k = (I-X)^{-1}$.
