If matrix $\sum_0^\infty C^k$ is convergent, how can I prove that $A(\sum_0^\infty C^k)B$ is convergent? For an $n \times n$ matrix $C$ and If $\sum_0^\infty C^k$ is convergent, how can I prove that for two matrices $A$ and $B$, $A(\sum_0^\infty C^k)B$ is convergent?
It seems quite obvious that you just multiply the convergent matrix sum with two constant matrices.
So, it has got to be convergent.
But I don't know how to prove it.. 

If the series converges to $M = \sum_{k=0}^{\infty} C^k$ then one has, using as norm the sum of squares (or sum of absolute values if it is a complex matrix) of the elements of the matrix:
$$
||A \sum_{k=0}^N C^k B - AMB|| = ||A(\sum_{k=0}^N C -M)B|| \leq ||A||||B|| \epsilon
$$
for $N$ large enough, from whence the convergence follows.
