数式雑記場3

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$\frac{1}{2}\cdot\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\ldots}}}}=\Box$

$\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\ldots}}}}$

$\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\ldots}}}}=x$

$\frac{1}{1+x}=x$

$x^2 + x - 1 = 0$

$x = \frac{-1\pm\sqrt{5}}{2}$

$x = \frac{\sqrt{5}-1}{2}$

$\frac{1}{2}\cdot\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\ldots}}}}=\frac{1}{2}\cdot\frac{\sqrt{5}-1}{2}=\frac{\sqrt{5}-1}{4}$

$x=a[1] + a[2] + \cdots + a[n]$

$\frac{1}{a[1]} + \frac{1}{a[2]} + \cdots + \frac{1}{a[n]} = 1$
$\{a[n]\}(x)$

$a[n] = (2n-1)^{2n-1}$
$\sum_{k=1}^{n+1}a[k]x^{n-k+1}=0$

$f(x) = \sum_{k=1}^{n+1}a[k]x^{n-k+1} = a_1x^n + a_2x^{n-1} + a_3x^{n-2} + \cdots + a_nx + a_{n+1}$

$f^{(n-2)}(x) = 0$

$f^{(n-2)}(x) = \frac{n!}{2!}a_1x^2 + (n-1)!a_2x + (n-2)!a_3 = 0$

$D = \{(n-1)!\}^2\cdot27^2 - 4\cdot\frac{n!}{2!}(n-2)!\cdot3125$
$=(n-1)!\{27^2(n-1)! - 6250n(n-2)!\}$
$=(n-1)!(n-2)!\{729(n-1) - 6250n\}$
$=(n-1)!(n-2)!(-5521n - 729n)$

$\frac{1}{p}(n+1)^p-\frac{1}{p}n^p-\frac{1}{p}$

$=\frac{1}{p}\{(n+1)^p-n^p-1\}$
$=\frac{1}{p}\{(_pC_0n^p + _pC_1n^{p-1} + _pC_2n^{p-2} + \cdots + _pC_{p-2}n^2 + _pC_{p-1}n^1 + _pC_pn^0)-n^p-1\}$
$=\frac{1}{p}\{(n^p + _pC_1n^{p-1} + _pC_2n^{p-2} + \cdots + _pC_{p-2}n^2 + _pC_{p-1}n^1 + 1)-n^p-1\}$
$=\frac{1}{p}(_pC_1n^{p-1} + _pC_2n^{p-2} + \cdots + _pC_{p-2}n^2 + _pC_{p-1}n^1)$