Accuracy, Fibonacci + Golden Ratio

[Gelsamel Epsilon]

I have been curious about this for a while...

I'm interested to know if there is any easy way to tell the accuracy of the (n+1)th on the nth term of the Fibonacci series in relation to the golden ratio.

I know that as n tends to infinity the ratio tends to the Golden Ratio "Phi" - but is there a way to tell, say, to how many decimal places the 32nd on the 31st term is close to Phi?

 

[Crosson]

We can do even better, and give Binet's closed-form expression for the $n^{th}$ Fibonacci number in terms of the golden ratio $\phi$:

$$F(n) = \frac{\phi^n - (1 - \phi)^n}{\sqrt{5}}$$

Sorry to give away so much, but mathematics is large enough :)