Let $\{A_k, k=0,1,\ldots\}$ be a sequence of complex square matrices such that $A_k - A_{k-1} = C^k$ where spectral radius of $C$ is less than 1. I want to show that $\lim_{k \to \infty} A_k$ exists. Is the following approach correct?
If we define $B_k = A_k - A_{k-1}$, then, we know $\lim_{k \to \infty} B_k =0$. Therefore, $\lim_{k \to \infty} a_k$ exists.
No. As others have stated, the standard example is $a_k =\sum_{j=1}^k \frac1{j} $ where $a_k-a_{k-1} =\frac1{k} \to 0$ but $a_k \to \infty$.
However, if $a_k-a_{k-1} = c^k$, then, summing from $1$ to $n$, $\sum_{k=1}^n(a_k-a_{k-1}) = \sum_{k=1}^nc^k$, or $a_n-a_0 =\dfrac{c-c^{n+1}}{1-c} =\dfrac{c}{1-c}-\dfrac{c^{n+1}}{1-c} $ so, since $c^{n+1} \to 0$, $a_n \to a_0+\dfrac{c}{1-c} $.