I've recently been doing some limits with circuits and such, and I came up with the following equation, $R$ being a constant:
$$f(x) = \frac{f(x-1)*R}{R+f(x-1)}+R$$
with $f(1)=2$.
I know that this series converges (because if I apply the limit test, it passes), and after using programming, I know that it converges to $(1 + \sqrt{4R+1})/2$ , but I do not know how to prove it.
Actually, it's $f(x)\to \frac{1+\sqrt{4R^2+1}}{2}$. You can use the Algebraic Limit Theorem (http://en.wikipedia.org/wiki/Algebraic_limit_theorem) to break down the limit and reduce this to the equation $L=\frac{L*R}{R+L}+R$, where $L=\lim_{x\to\infty} f(x)$, and solve using the quadratic formula, throwing out the positive solution.
Note that this only works for positive $R$.