How to write $r(a) = \lim_{n\to∞}r_n(a)$ as an infinite product?

Question

Define $$\begin{align}r_1&:\mathbb N\to\mathbb R, a\to\sqrt{a} \\ r_{n+1}&:\mathbb N\to\mathbb R, a\to\sqrt{a+r_n(a)}&\forall n\in\mathbb N\end{align}$$ and $r(a) = \lim\limits_{n\to\infty}r_n(a)$. I want to write $r(a)$ as an infinite product, but I don't even know where to begin. If it helps I have worked out the limit $$r(a) = \frac {1+\sqrt{1+4a}}2$$

Answer 1

This is an infinite product:

\begin{align} r(a)&=\sqrt a \times \frac{\sqrt{a+\sqrt a}}{\sqrt a} \times \frac{\sqrt{a+\sqrt{a+\sqrt a}}}{\sqrt{a+ \sqrt a}}\times\dots\\ &=\prod_{j=1}^\infty s_j(a) \end{align}

where $s_1(a)=\sqrt a$ and $s_j$ is defined as $r_j/r_{j-1}$ for $j\ge 2$, according to the iterative definition given for $r_n$ in your question.