query: Limit of Ratio of Adjacent Fibonacci numbers $\to \phi$
positive_passages.docid: 14129
positive_passages.text: How to prove that $\lim \limits_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}$
negative_passages.docid: 148492
negative_passages.text: Finding the limit of the sequence $(1 - 1/\sqrt 2) \dotsm ( 1 - 1/\sqrt {n+1})$
negative_passages.docid: 33144
negative_passages.text: Criterion for $\phi(mn)=m\phi(n)$ to be true
negative_passages.docid: 119891
negative_passages.text: Divisibility and the Fibonacci sequence
negative_passages.docid: 117681
negative_passages.text: For which Natural $n\ge2: \phi(n)=n/2$
negative_passages.docid: 34025
negative_passages.text: Limit of the Sequence: $a_{n+1}=a_n+\frac{1}{a_n}$ with $a_1=1$
negative_passages.docid: 26973
negative_passages.text: Relation between Pascal's triangle and fibonacci series.
negative_passages.docid: 145255
negative_passages.text: Fibonacci, tribonacci and other similar sequences
negative_passages.docid: 88662
negative_passages.text: $n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer.
negative_passages.docid: 43865
negative_passages.text: Find all positive integers $n$ such that $\phi(n)=6$.
negative_passages.docid: 12031
negative_passages.text: Asymptotic behavior of the sequence $a_n=\sqrt{n+a_{n-1}}$
negative_passages.docid: 30273
negative_passages.text: Fibonacci Recurrence Relations
negative_passages.docid: 5797
negative_passages.text: Limit of the sequence $a_n=\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}$
negative_passages.docid: 102425
negative_passages.text: Limit $ a_{n+1} = \frac{1}{3}(2a_n + \frac{5}{a_n^{2}}), a_1 > 0 $
negative_passages.docid: 29609
negative_passages.text: Limits involving factorials $\lim_{N\to\infty} \frac{N!}{(N-k)!N^{k}}$
negative_passages.docid: 4512
negative_passages.text: Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?
negative_passages.docid: 67420
negative_passages.text: Limit of quotient sequence