Given two convergent sequences $\{x_k\}$ and $\{y_k\}$, with rates of convergence $r_1$ and $r_2$. It is known that $\{x_k y_k\}$ is also convergent. In addition to the convergence, can we get the rate of convergence of $\{x_k y_k\}$?
We say that the sequence $\{x_k\}$ converges with order $q$ to $L$ if
$$\lim_{k \rightarrow \infty }\frac {|x_{k+1}-L|}{|x_{k}-L|^{q}}<M.$$