Goal:
Find $$\lim_{n \rightarrow \infty} d_{n+1}^2$$ where $$d_{n+1}^2 = d_n^2 + r^2 - 2 \cdot r \cdot d_n \cdot \cos \theta$$ for constant $r$ and $\theta < \frac{\pi}{2}$.
Attempt:
\begin{align*} a &= \lim_{n \rightarrow \infty} d_{n+1}^2 \\ &= \lim_{n \rightarrow \infty} d_n^2 + r^2 - 2 d_n r \cos \theta \\ &= a + r^2 - 2 r \cos \theta \sqrt{a} \\ &= \frac{r^2}{4 \cdot \cos^2 \theta} \end{align*}
Questions:
I know this is the correct answer from running simulations but I'm not confident in the limit approach I'm using. First of all, I know that the limit diverges for $\theta \geq \frac{\pi}{2}$ but I never (knowingly) use the fact that $\theta < \frac{\pi}{2}$ in the proof. So what is going on here? Is there a different approach I should take?
Your method works perfectly only if the limit exists. If the limit doesn't exist, then this approach can give you an answer even though the limit doesn't make sense. So you've implicitly used $\theta<\frac{\pi}{2}$ in assuming that $a$ makes sense.