what is the "form" of the following recursion relation?

bjnartowt

Hi all, I have a recursion relation I am trying to solve:

[itex]{X_n} = \frac{1}{{1 - {\alpha _0} \cdot {X_{n - 1}}}} \to {X_n} = ?[/itex]

What is the "mathematical form" of this recursion-relation? E.g., I know what a homogeneous, linear recursion-relation with constant coefficients looks like, and how to solve it; same with an inhomogeneous recursion relation. But what about this one? (alpha0 = a constant). All I know is that it looks like the closed-form solution to the infinite geometric sum, and I don't know where to go from there. If someone tells me what the mathematical form of this is, I can Google example-solutions that I can work off of, and/or see what a textbook says.


bjn

JJacquelin

Hi !

In attachment, you can see the method (not the whole calculus) which leads to the closed form.

Attached Thumbnails

 

bjnartowt

Thanks, JJaquelin. As it turns out, I found a book on difference-equations, in which there is a topic on continued fractions, which is a less mainstream topic than differential equations, so I am reading that now.

FYI: the book is Elaydi Saber's book: "An Introduction to Difference Equations".