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Thread: Golden ratio representation

  1. #1 Golden ratio representation 
    Moderator Moderator AlexP's Avatar
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    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

    .


    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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  3. #2  
    Forum Ph.D.
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    If you let then which you can then solve as a quadratic in .


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  4. #3  
    Moderator Moderator AlexP's Avatar
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    ok, I get it. lot simpler than I imagined it would be. thanks.
    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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  5. #4  
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    Is there anything more amazing, literally amazing, than the Fibonacci thing, in math? (In nature)

    If there is, I want to know about it.

    Though, I'm pretty amazed-enough, already.
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  6. #5  
    Forum Professor river_rat's Avatar
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    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.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  7. #6  
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    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?
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  8. #7  
    Forum Professor river_rat's Avatar
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    Ah you see accountabled, you first need to check if your second continued fraction converges...
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  9. #8  
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    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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