Let $b$ and $a_0$ be real numbers and $(a_n)$ a sequence such that: $$a_{n+1} = \frac{1}{1+ba_n}$$ For what values of $b$ and $a_0$ does this sequence converge, and to what number does it converge to?
This seems to be a common problem given to Calculus students, given $b=2$ and $a_0=1$. In that case, it converges to $\frac12$. A friend of mine is wondering about the general case given above, so I'm asking here. We have been able to make a few observations:
- It is undefined if $a_0 b = -1$.
- It seems to converge for positive values of $b$ and $a_0$.
- It seems to diverge if either $b$ is negative, no matter the $a_0$.
- With $b$ fixed at $2$ and $a_0=-1$, the sequence converges to $-1$. Is there any other $a_0$ such that also happens?
We haven't really been able to prove any of the above except the first, probably because we're not used to dealing with the convergence of recursive sequences specifically, so some help would be appreciated.