and \(\mathbf{z}_m=R_{\theta_m}\cdot\mathbf{z}_{m-1}\), where \(R_{\theta}\) is the rotation matrix.

Further, \(\varphi_m=\cot^{-1}\frac{x_m}{y_m}\).  Thus, we are evaluating \(\lim_{n\rightarrow\infty}\cot^{-1}\frac{x_n}{y_n}=\cot^{-1}\ \left(\lim_{n\rightarrow\infty}\frac{x_n}{y_n}\right)\).  Further, the above recursion relations for   \(\cos\varphi_m,\ \sin\varphi_m\) translate into the following recurrence relation for the vector