Suppose we have a sequence $\{x_k\} \subset X$ with $X$ being a Banach space satisfying $$\| x_k - x_{k-1} \| \le \frac{1}{k^2} C,$$ where $C$ is some positive constant. The sequence clearly has a limit since for sufficiently large $n, m$ \begin{align*} \|x_n - x_m\| \le \sum_{j=n}^m \|x_j - x_{j+1}\| \le \sum_{j=n}^m \frac{C}{j^2} \le \varepsilon, \end{align*} for every $\varepsilon > 0$.
I am wondering whether we could determine the rate of convergence for the sequence $\{ \|x_k - x\| \}$ as a function of $k$ where $x$ is the limit point? It seems triangle inequality does not give a sensible result.