What is the form of the following recursion relation?

In summary, The conversation is about a recursion relation that the speaker is trying to solve and is asking for the mathematical form of the relation. The speaker mentions that they know how to solve certain types of recursion relations, but they are unsure about this one. They also mention that the closed-form solution looks similar to an infinite geometric sum. Someone replies with a method for finding the closed-form solution and the speaker mentions a book they found on the topic.
  • #1
bjnartowt
284
3
what is the "form" of the following recursion relation?

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
 
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  • #2


Hi !

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

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  • #3


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".
 

FAQ: What is the form of the following recursion relation?

What is a recursion relation?

A recursion relation is a mathematical equation that defines a sequence of values based on previous terms in the sequence. It is a way to express a problem in terms of its subproblems.

What are the benefits of using a recursion relation?

Recursion relations allow for efficient and concise solutions to problems that can be broken down into smaller subproblems. They also provide a more elegant and intuitive approach to problem-solving in some cases.

Can you give an example of a recursion relation?

One example of a recursion relation is the Fibonacci sequence, where each term is the sum of the two previous terms. This can be expressed as the relation f(n) = f(n-1) + f(n-2), with the initial values of f(1) = 1 and f(2) = 1.

What is the form of a recursion relation?

The form of a recursion relation is typically expressed as a function or equation, with one or more variables and a base case or initial condition. It may also involve a recursive call, where the function calls itself with different parameters.

How is a recursion relation different from an iterative solution?

A recursion relation and an iterative solution both involve repetition, but a recursion relation breaks the problem down into smaller subproblems, while an iterative solution uses a loop to repeatedly perform a specific set of instructions.

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