It is relying on a theorem which is not so easy to show that all continued fractions converge (the ones with integers).
So we know that
Converges to some number.
But,
For it repeats.
Thus,
Thus,
Thus,
Thus,
This tells us that this is one of these two possible values.
It cannot be negative thus,
The Divine Proportion (My body is shaped in Divine Proportion).
Hello, Cilia!
This is a vast and intricate topic which could take months to explain.
If you know the very basics, I can show you a tiny sliver of the whole sprectrum.
Your example:
By inspection ("eyeballing" it),
. . we see that each term is: one plue one over the preceding term.
That is: .
Assuming that the fraction goes on forever: .
. . can we determine its value?
Hence, we have: .
Quadratic Formula: .
Since is positive, we have: .
. . which happens to be the Golden Mean, .
A useful link: Continued fraction - Wikipedia, the free encyclopedia