Convergence of sequence of vectors in $C^*$-algebra

Question

Let $B$ is $C^*$-algebra and $x_i \in B$ - linear independent vector system, $\alpha_i \in \mathbb{C}$ such that: $$\|x_i\| = 1$$ $$\lim_{N \to \infty} \|\sum_{k=1}^N \alpha_k x_k\| = \lim_{N \to \infty} \sum_{k=1}^N \alpha_k = C < \infty$$ is it true, that $\sum_{k=1}^\infty \alpha_k x_k$ converge?