This question was part of my previous year mid term exam of Linear Algbera and I am really struck on it.

Let A be $n times m $ matrix with real entries, and let $B=AA^t$ and let $alpha$ be the supremum of $x^t Bx$ where supremum is taken over all vectors $xin mathbb{R}^n$ with norm less than or equal to 1. Consider $C_k = I + sum_{j=1}^{k} B^j$. Show that the sequence of matrices $C_k$ converges iff $alpha <1$.

The problem is that I am not even able to start.

I have been following Hoffman and Kunze linear algebra and I am doing a graduate level course in it.

Any help would be deeply appreciated.