I have to solve a very strange recurrence relation. Has anyone an idea how to derive a closed formula for $r_i$? At least a 'good' upper bound would be nice. Currently I have no idea how to solve this...
Thanks for your help.
For the recurrence, $n \in \mathbb{N}$.
\begin{align*} r_0 &= \frac{n}{2\pi} \\ r_i &= r_{i-1} - \frac{\sin \left(\frac{\pi\cdot(n-2)}{2n} + \arcsin \left (\frac{2\pi\cdot r_{i-1}}{n} \cdot \sin \left( \frac{\pi\cdot(n-2)}{2n}\right) \right) \right)}{2 \cdot \sin\left( \frac{\pi\cdot(n-2)}{2n}\right)} \end{align*}