Analytical convergence of an equation?

Question

I am reading a paper which shows the following equation:

$$ x_t = x_i + \alpha x_{t-1} $$

Where $x_i$ is the initial value, and $\alpha$ is a simple real constant.

I am trying to ascertain if there is a way for me to (analytically) determine the convergence of $x_t$ if I am given $\alpha$ and $x_i$? I want to do this because I believe there is an error in the paper.

Thanks!

Answer 1

Hint:   if $\,\alpha=1\,$ then $\,x_t\,$ is an arithmetic progression, otherwise $\,x_t\,$ differs by a constant from a geometric progression with common ratio $\,\alpha\,$, since:

$$ x_t = x_i + \alpha x_{t-1} \quad\iff\quad x_t + \dfrac{x_i}{\alpha-1} = \alpha \left(x_{t-1}+\dfrac{x_i}{\alpha-1}\right) $$