If a recursive sequence converges, must its inverse be divergent? Suppose I have a recursive sequence $\displaystyle a_{n+1} = \frac{a_{n}}{2}$. Clearly, the sequence converges towards zero. Now, suppose I define an "inverse" sequence $\displaystyle b_{n+1} = 2b_{n}$. While the sequence clearly diverges, could I know that based on the knowledge that $a_{n+1}$ converges? Is there any sequence that converges, whose "inverse" also converges?
.. and could this "inverse" sequence be defined in a more intuitive way?

Consider the sequence $a_{k+1} = \sinh(a_k)$. If started at $a_0=\exp(i \pi/4)$ then this converges to $\lim_{n \to \infty} a_n=0$.
Now look at the inverse $ a_{k-1} = \operatorname{asinh}(a_k)$. you can step backwards to arrive at $a_0$ again and go to $\lim_{n \to -\infty} a_n=0$ 
When I came across it this surprised me much. See a picture here: 
