Let $\{u_k\}_{k\in \mathbb{N}}$ and $\{v_k\}_{k\in \mathbb{N}}$ be the two sequences given by $$ \begin{cases} u_0=v_0=1,&\\ u_{k+1}=\left(1-\frac{1}{(k+1)^2}\right)u_k+\left(\frac{1}{k+1}\right)v_k, & (k\in \mathbb{N})\\ v_{k+1}=\left(-\frac{1}{k+1}\right)u_k+\left(1-\frac{1}{(k+1)^2}\right)v_k. & (k\in \mathbb{N}) \end{cases} $$ Checking the convergence of the sequences $\{u_k\}_{k\in \mathbb{N}}$ and $\{v_k\}_{k\in \mathbb{N}}$.
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Let $z_k = u_k + i v_k$. The recursion says $z_{k+1} = a_{k+1} z_k$, where $a_k = 1 - 1/k^2 - i/k$. Now $a_k = r_k e^{i\theta_k}$ where $r_k = |a_k| = \sqrt{1 - 1/k^2 + 1/k^4}$ and $\theta_k = -\arctan\left(\frac{1/k}{1 - 1/k^2}\right)$. Thus $z_n = z_0 \prod_{k=1}^n a_k = \left(\prod_{k=1}^n r_k \right) \exp\left(i \sum_{k=1}^n \theta_k\right)$. We have $r_k = 1 - 1/(2 k^2) + O(1/k^4)$ and $\theta_k = -1/k + O(1/k^3)$ as $k \to \infty$. The infinite product $\prod_{k=1}^\infty r_k$ converges to a nonzero limit $R$, so $|z_n| \to R |z_0|$ as $n \to \infty$, but $\sum_{k=1}^\infty \theta_k$ diverges, so $\arg(z_n)$ diverges, and so do the sequences $u_n$ and $v_n$. In the limit as $n \to \infty$, the points $z_n$ spiral infinitely many times around the circle of radius $R |z_0|$ centred at $0$. |
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