I need to show that the following series is always convergent: $$\left|\left|v\right|\right|=\sum_{k=0}^{\infty}\frac{\left|\mathcal{A}^kv\right|}{R^k},$$
where $\mathcal{A}$ is a linear operator and $R>\rho\left(\mathcal{A}\right)=\underset{i}{max}|\lambda_i|$. The R.H.S. of the inequality is the spectral radius of the linear operator $\mathcal{A}$.
I know that i need to show that there exists some $C$ such that $\left|\left|\mathcal{A}\right|\right|_2\leq Cr^n$. Thereafter i propably need to show that the sum is bounded by some series involving the above mentioned $C$.
But i cant seem to do so. How do i go about this?
Take $r$ so $\rho(A) < r < R$. By Gelfand's spectral radius formula, $\|A^k\|^{1/k} < r$ for sufficiently large $k$, where $\|\cdot\|$ is the operator norm induced by the norm $|\cdot |$, and then $|A^k v|/R^k \le \|A^k\| |v|/R^k < (r/R)^k |v|$ so the series converges.