Originally Posted by

**hollywood** If you look at the original post, he's really talking about sequences, not series.

There's not a whole lot of difference between taking the limit of a sequence and a function. Since you know limits for functions, you should be able to evaluate $\displaystyle \lim_{n\to\infty}x^n$ in the cases $\displaystyle 0\le{x}<1$, $\displaystyle x=1$, and $\displaystyle x>1$. When x is negative, there are some difficulties with the definition of $\displaystyle x^n$, but since you know n is an integer, it just breaks up into the two cases n odd or n even.

For the Fibonacci sequence, you can prove the formula $\displaystyle F_n = \frac{1}{\sqrt{5}}(\phi^n-(-\phi)^{-n})$ by just checking the first two terms and the recurrence relation. Then you can see that $\displaystyle \lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\phi$.

- Hollywood