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Fungrim entry: d56025

limnFn+mFn=φm\lim_{n \to \infty} \frac{F_{n + m}}{F_{n}} = {\varphi}^{m}
Assumptions:mZm \in \mathbb{Z}
TeX:
\lim_{n \to \infty} \frac{F_{n + m}}{F_{n}} = {\varphi}^{m}

m \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
SequenceLimitlimnaf(n)\lim_{n \to a} f(n) Limiting value of sequence
FibonacciFnF_{n} Fibonacci number
Infinity\infty Positive infinity
Powab{a}^{b} Power
GoldenRatioφ\varphi The golden ratio (1.618...)
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("d56025"),
    Formula(Equal(SequenceLimit(Div(Fibonacci(Add(n, m)), Fibonacci(n)), For(n, Infinity)), Pow(GoldenRatio, m))),
    Variables(m),
    Assumptions(Element(m, ZZ)))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-01-31 18:09:28.494564 UTC