Fibonacci sequence: Difference between revisions

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=== Limit of consecutive quotients ===

[[Johannes Kepler]] observed that the ratio of consecutive Fibonacci numbers [[convergent sequence|converges]]. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio <math>{{tmath|\varphi </math>; }}:<ref>{{Citation|last=Kepler |first=Johannes |title=A New Year Gift: On Hexagonal Snow |year=1966 |isbn=978-0-19-858120-8 |publisher=Oxford University Press |page= 92}}</ref><ref>{{Citation | title = Strena seu de Nive Sexangula | year = 1611}}</ref>

<math display=block>\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\varphi.</math>