Ratio of two successive Fibonacci number is Golden Ratio (φ)
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- เผยแพร่เมื่อ 5 ก.พ. 2025
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Using limit we will prove that the ratio of two successive Fibonacci numbers is approaching to Golden Ratio = φ
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Thank you
Fantastic!
Nicely done
How you prove that the limit exists?
Normally, to prove if limit exists we check whether left limit exists and right limit exists and they are equal, more formally to determine if the limit of 𝑓 ( 𝑥 ) at 𝑥 = 𝑎 exists, we check three things: if the left limit of 𝑓 ( 𝑥 ) at 𝑥 = 𝑎 exists, if the right limit of 𝑓 ( 𝑥 ) at 𝑥 = 𝑎 exists, if these two limits are equal. In this specific limit since we can't.check left and right side of infinity ♾️, as it grows fast and dynamically, we can attempt to use law of large numbers and see where the limit is going. In addition we check Horizontal and Vertical asymptotes of the function using limit definition such that lim as x is approaching to positive or negative ♾️, if y is approaching to a finite value c ( converging) that means we have H.A. at y=c or as x is approaching to positive or negative infinity , if y value is also approaching to positive or negative infinity ( diverging) then we can conclude we have V.A. intuitively or by showing some steps on table we can deduce that function is converging or unbounded. In limit of Fibonacci series we see that it converges to a certain value : golden ratio. It has asymptote at y=φ
@@mathbyleo yes I agree. But I'm saying that proving that the limit exists in this case is not that easy. Once you prove that the limit exists, then your proof is absolutely fine.
Maybe we need to dig deep into existence of end of irrational numbers ?! What is your idea towards ?
There is one proof using the Binet's formula. But I was looking for some other proof considering sequences only.