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.. 

One approach is to note that the map
$$
X \mapsto AXB
$$
is continuous.  As Marc van Leeuwen notes below, 
$$
A(\sum_{k=0}^N C^k)B = \sum_{k=0}^N AC^kB
$$
so that the limit must be $\sum_{k=0}^\infty AC^kB$, which converges (by our statement of continuity).
