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  • 02:0102:01, 1 March 2024 diff hist +94 m User:RandomUndergrad/SandboxNo edit summary current
  • 01:5901:59, 1 March 2024 diff hist +1,595 N User:RandomUndergrad/Sandbox/Bounds to Recursive FunctionsCreated page with "== Binet First Form == Consider $U_n = mU_{n-1} + U_{n-2}, U_0 = 0, U_1 = 1$, with $\Delta = \sqrt{m^2+4}, \alpha = \frac {m+\Delta}2, \beta = \frac {m-\Delta}2$. Then $U_n = \frac {\alpha^n - \beta^n}\Delta = \frac {\alpha^n - \beta^n}{\alpha - \beta} = \alpha^{n-1} + \alpha^{n-2}\beta + \dots + \beta^{n-1}$. Also $m\alpha^{n-2} \le U_n \le \alpha^{n-1}$, and $|m\beta^{2-n}| \le U_n \le |\beta^{1-n}|$ (at least for $m > 0$.) == Proof == Note that $\alpha \beta = \..." current

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