TITLE: Solve $\alpha_{n+1}=-1/\alpha_n+\beta_n$, with $\lim_{n\rightarrow\infty}\alpha_n=0$.
QUESTION [0 upvotes]: Suppose that $(\beta_n)_{n\in\mathbb{N}}$ is a strictly positive, real-valued sequence and is known.
Can we find a $[0,\infty]$-valued sequence $(\alpha_n)_{n\in\mathbb{N}}$ satisfying the following recursion?
$$\alpha_{n+1}=-1/\alpha_n+\beta_n,\quad n\in\mathbb{N};$$
with boundary condition
$$\lim_{n\rightarrow\infty}\alpha_n=0.
$$
I have no idea how to proceed with this one. Any help much appreciated!

REPLY [2 votes]: Generally, no. 
We have $\beta_n = \alpha_{n+1} + \frac1{\alpha_n}$ and with $\lim_{n\to\infty}\alpha_n = 0$ and $\alpha_n > 0$ this implies $\lim_{n\to\infty}\beta_n = \infty$.
So the given series $(\beta_n)$ converging to $\infty$ is a necessary condition for the $(\alpha_n)$ to exist.
I have no idea if it is also sufficient, I suspect not.
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