First I found the value of $x_n$ when $n$ goes to infinity.
$$\lim_{n \to \infty} \dfrac{1}{n} = 0 = a.$$
Then I want to find $c$ and $\rho$ such that $$|x_{n+1} -a| = c|x_{n} - a|^\rho$$
But I get stuck with this step.
Can anyone offer some assistance, please?
Let $\{x_n\}_{n=1}^\infty \subset \mathbb{R}$ be a strictly positive sequence which converges to zero. We say that that the order of convergence is $p \ge 1$ and the rate is $c > 0$ provided $$ \frac{x_{n+1}}{x_n^p} \rightarrow c, \quad n \rightarrow \infty, \quad n \in \mathbb{N}.$$ If $p = 1$, then we require $c \in (0,1)$. The case of $p=1$ is called linear convergence, the case of $p=2$ is called quadratic convergence, the case of $p=3$ is called cubic convergence. Your sequence given by $$x_n = \frac{1}{n}$$ is a corner case, because $$ \frac{x_{n+1}}{x_n} = \frac{n}{n+1} \rightarrow 1, \quad n \rightarrow \infty, \quad n \in \mathbb{N}.$$ This is called sublinear convergence.