1. One representation of the golden ratio, , is

I'm curious, how do you actually calculate something like this?

Another example of an interesting infinite square root thing is

.

2.

3. If you let then which you can then solve as a quadratic in .

4. ok, I get it. lot simpler than I imagined it would be. thanks.

5. Is there anything more amazing, literally amazing, than the Fibonacci thing, in math? (In nature)

If there is, I want to know about it.

6. Just as a general warning - Jane's method works if the limit exists. It does not show that the limit exists though. This caveat has got many people into trouble in the past.

7. Would this also be valid for complex limits?

E.g.

Is the limit of the series:

And this equals:

which is the solution to: (analogous to: for the Golden Ratio).

Another way of representing the Golden Ratio is:

but we could also try:

which has a limit of : (difficult to calculate but easy to derive from the recurrence relation (for n=1).

Or is this too naughty mathematics?

8. Ah you see accountabled, you first need to check if your second continued fraction converges...

9. Well, it obviously doesn't seem to converge at first sight:

soon leads to a division by zero.

Also applying a trick like injecting an integer doesn't seem to help:

since it doesn't seem to converge for any integer we plug in.

But, the series:

does nicely converge to:

Or am I violating a mathematical rule here?

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