0
$\begingroup$

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!

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.