Let $E$ be a Banach Algebra with identity, and $v\in E$, so that $||v|| < 1$. The geometric series $w = \sum_{k=0}^\infty v^k$ converges in the norm.
I can show that $||w|| \le \frac{1}{1-||v||}$, but does the equality hold? If it holds, how can I show this? If it doesn't hold, are there counterexamples?