sequence with a quadratic form of matrices I just face to a sequence with a quadratic form of matrices, when I work on discrete Lyapunov equation:
$$\lim_{n\to\infty}\sum_{k=0}^n A^kR_0(A^*)^k = R_0+AR_0A^*+A^2R_0(A^*)^2+...+A^{n-1}R_0(A^*)^{n-1}+A^nR_0(A^*)^n$$
The matrix A is a stable matrix (all its eigenvalues are in the unit disk) and $R_0$ could be an arbitrary matrix (the relaxed version could be the block diagonal case). I did so many searches and didn't find any special answers to these questions:
1) Condition(s) of convergence
2) The amount of limit if the sequence is convergence.

If $A$ is stable then the spectral radius is less than one and the series certainly converges absolutely (it follows from Gelfand's formula which gives estimates on $\|A^n\|$).
Let's calculate
$$
P=\sum_{k=0}^\infty A^kR_0(A^*)^k=R_0+\sum_{k=1}^\infty A^kR_0(A^*)^k=
R_0+A\cdot \underbrace{\sum_{k=1}^\infty A^{k-1}R_0(A^*)^{k-1}}_{=P}\cdot A^*.
$$ 
Therefore, the sum $P$ is the solution to the Lyapunov equation
$$
APA^*-P+R_0=0.
$$
