피보나치 수열

간단한 요약

• 정의
  
  • \(F_0=0, F_1=1\)
  • \(F_{n+2}=F_{n+1}+F_{n}\)
• 잘 알려진 성질들
  
  • \(\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887\cdots\)[1][2]
• \(\lim_{n\to\infty}\frac{F(n+1)}{F(n)}=\varphi\)
•  
  http://bomber0.byus.net/mimetex/mimetex.cgi? (-1)^{n} = F_{n+1}F_{n-1} - F_n^2\
  
• \( F_{n+1}F_{n-1} - F_n^2=(-1)^{n}\)
•  
  http://bomber0.byus.net/mimetex/mimetex.cgi?\Sigma_{n=1}^{\infty}\frac{(-1)^{n+1}}{F_nF_{n+1}}=\Sigma_n^{\infty}\frac{F_{n}}{F_{n+1}}-\frac{F_{n-1}}{F_{n}}}=\varphi -1
  
• \(\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{F_nF_{n+1}}=\sum_{n=1}^{\infty}{\frac{F_{n}}{F_{n+1}}-\frac{F_{n-1}}{F_{n}}}=\varphi -1\)
•  
  http://bomber0.byus.net/mimetex/mimetex.cgi?\Prod_{n=2}^{\infty}(1+\frac{(-1)^{n}}{F_n^2})=\Prod_{n=2}^{\infty}\frac{F_n^2+(-1)^n}{F_n^2}=\Prod_{n=2}^{\infty}\frac{F_{n-1}F_{n+1}}{F_n^2}=\varphi
  
• \(\prod_{n=2}^{\infty}(1+\frac{(-1)^{n}}{F_n^2})=\prod_{n=2}^{\infty}\frac{F_n^2+(-1)^n}{F_n^2}=\prod_{n=2}^{\infty}\frac{F_{n-1}}{F_n}\frac{F_{n+1}}{F_n}=\varphi\)