Insights Reduction of Order For Recursions

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This is not meant as a full introduction to recursion relations but it should suffice for just about any level of the student.
Most of us remember recursion relations from secondary school. We start with a number, say, 1. Then we add 3. That gives us 4. Now we that number and add 3 again and get 7. And so on. This process creates a series of numbers ##\{ 1, 4, 7, 10, \dots \}##. Once we get beyond that stage we start talking about how to represent these. Well, let’s call the starting number ##a_0##. Then the next number in the series would be ##a_1##, then ##a_2##, …, on to ##a_n## where n is the nth number in the series. We may now express our recursion as a rule: ##a_{n +1} = a_n + 3##, where ##a_0 = 1##.
We may go much further. We can talk about things like the Fibonacci series: ##F_{n + 2} = F_{n + 1} + F_{n}##, where ##F_1 = F_2 = 1##. This is the famous series ##\{ 1, 1, 2, 3, 5, 8, 13, \dots \}##. But what if we want a formula to find out what ##F_n## is for the nth number...

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topsquark said:
Is the term "recursion relation" in common use? I would call it a "recurrence relation".

Also I would say "finite difference calculus" instead of "difference calculus".
 
pbuk said:
Is the term "recursion relation" in common use? I would call it a "recurrence relation".

Also I would say "finite difference calculus" instead of "difference calculus".
Sorry! As I understand it both terms (the recursive and finite difference) are in use. However, the only work I've done with recurrence relations comes from High School and I couldn't really find much on the web about finite difference calculus. I had to fill in a lot of gaps to do this, so I may not be using standard terminology.

-Dan
 
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Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.

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