I'm stuck on a problem in my calculus-class. My first task is to find the general solution to the recurrence relation: $\ x_{n+2}-5x_{n+1}-36x_n$ = 0
This I've found to be: $x_n = C9^n+D(-4)^n$.
But then the next task is: Let $(x_n)^\infty_1$ be an arbitrary solution to the recurrence relation above, that isn't the solution $x_n = 0$. Find the limit:
$\lim_{n\to\infty} (x_{n+1}/x_n)$
for the different possibilities you have found above.
And this is where I'm stuck. I have no idea how to approach this problem, so any pointers would be very much appreciated. If something is unclear it is probably because I've translated this from another language, so please ask if something doesn't make sense to you.
Okay, so you have $x_n = C9^n+D(-4)^n$ where $C$ and $D$ are not both $0$. We can put this directly in the limit: $$\lim_{n\to\infty}\frac{x_{n+1}}{x_n} = \lim_{n\to\infty}\frac{C9^{n+1}+D(-4)^{n+1}}{C9^n+D(-4)^n}.$$ If $C \neq 0$, then you can see that the $9^n$ will dominate; try dividing the top and bottom by $9^n$. On the other hand, if $C=0$, then the fraction will simplify by itself and you'll get a different result.