\\(n\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3059\u308b\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\\(k\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3059\u308b. \u95a2\u6570 \\((1-x)^n x^k\\) \u306e \\(0 \\leqq x \\leqq 1\\) \u306b\u304a\u3051\u308b\u6700\u5927\u5024\u3092 \\(a _ n\\) \u3068\u3059\u308b. \\(a _ n\\) \u304a\u3088\u3073 \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} a _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(f(x) , g(x)\\) \u3092 \\(0 \\leqq x \\leqq 1\\) \u306b\u304a\u3044\u3066\u5b9a\u3081\u3089\u308c\u305f\u9023\u7d9a\u95a2\u6570\u3068\u3059\u308b. \u95a2\u6570 \\((1-x)^n f(x)\\) , \\((1-x)^n g(x)\\) , \\((1-x)^n \\left\\{ f(x) +g(x) \\right\\}\\) \u306e \\(0 \\leqq x \\leqq 1\\) \u306b\u304a\u3051\u308b\u6700\u5927\u5024\u3092\u305d\u308c\u305e\u308c \\(b _ n , c _ n , d _ n\\) \u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \\(0 , b _ n +c _ n , d _ n\\) \u306e\u5927\u5c0f\u3092\r\n\\[\r\n\\square \\leqq \\square \\leqq \\square\r\n\\]\r\n\u306e\u5f62\u5f0f\u3067\u7b54\u3048, \u305d\u306e\u7406\u7531\u3092\u8ff0\u3079\u3088.<\/p><\/li>\r\n (3)<\/strong>\u3000\\(p , q , r \\geqq 0\\) \u3092\u5b9a\u6570, \\(f(x) = px^2+qx+r\\) \u3068\u3057, \u95a2\u6570 \\((1-x)^n f(x)\\) \u306e \\(0\\leq x \\leqq 1\\) \u306b\u304a\u3051\u308b\u6700\u5927\u5024\u3092 \\(e _ n\\) \u3068\u3059\u308b. \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} e _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(y = (1-x)^n x^k\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{align}\r\ny' & = -n (1-x)^{n-1} x^k +k (1-x)^n x^{k-1} \\\\\r\n& = -\\left\\{ (n+k) x -k \\right\\} (1-x)^{n-1} x^{k-1}\r\n\\end{align}\\]\r\n\\(0 \\lt x \\lt 1\\) \u306e\u7bc4\u56f2\u3067 \\(y'=0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nx = \\dfrac{k}{n+k}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(0 \\leqq x \\leqq 1\\) \u306b\u304a\u3051\u308b \\(y\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} x & 0 & \\cdots & \\dfrac{k}{n+k} & \\cdots & 1 \\\\ \\hline y' & & + & 0 & - & \\\\ \\hline y & 0 & \\nearrow & \\text{\u6700\u5927} & \\searrow & 0 \\end{array}\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\na _ n & = \\left( 1 -\\dfrac{k}{n+k} \\right)^n \\left( \\dfrac{k}{n+k} \\right)^k \\\\\r\n& = \\underline{\\dfrac{k^n n^k}{(n+k)^{n+k}}}\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\na _ n & = \\dfrac{1}{\\left( 1 +\\frac{k}{n} \\right)^{\\frac{n}{k} \\cdot k}} \\left( \\dfrac{k}{n+k} \\right)^k \\\\\r\n& \\rightarrow \\dfrac{1}{e^k} \\cdot 0 = 0 \\quad ( \\ n \\rightarrow \\infty \\text{\u306e\u3068\u304d} )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} a _ n = \\underline{0}\r\n\\]\r\n (2)<\/strong><\/p>\r\n \\(u(x) = (1-x)^n f(x)\\) , \\(v(x) = (1-x)^n g(x)\\) , \\(w(x) = u(x) +v(x)\\) \u3068\u304a\u304f. (3)<\/strong><\/p>\r\n \\(r (1-x)^n\\) \u306f, \\(0 \\leqq x \\leqq 1\\) \u306b\u304a\u3044\u3066, \\(x=0\\) \u306e\u3068\u304d\u306b, \u6700\u5927\u5024 \\(r\\) \u3092\u3068\u308b. \uff08 \\(n\\) \u306e\u5024\u306b\u3088\u3089\u306a\u3044. \uff09
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n\\(0 \\leqq x \\leqq 1\\) \u306b\u304a\u3044\u3066 \\(f(x) , g(x)\\) \u304c\u5b9a\u7fa9\u3067\u304d\u308b\u306e\u3067\r\n\\[\r\nu(1) = v(1) = w(1) = 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\nd _ n \\geqq 0\r\n\\]\r\n\u307e\u305f, \\(u(x) , v(x) , w(x)\\) \u304c, \u305d\u308c\u305e\u308c \\(x = b , c , d\\) \u306e\u3068\u304d\u306b\u6700\u5927\u5024\u3092\u3068\u308b\u3068\u3059\u308c\u3070\r\n\\[\\begin{align}\r\nb _ n +c _ n & = u(b) +v(c) \\\\\r\n& \\geqq u(d) +v(d) \\\\\r\n& = w(d) = d _ n\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\underline{0 \\leqq d _ n \\leqq b _ n +c _ n}\r\n\\]\r\n
\r\n\u307e\u305f, \\(u(x) = px^2 (1-x)^n\\) , \\(v(x) = px (1-x)^n\\) \u3068\u307f\u306a\u3057\u3066, \\(u _ (x) , v(x) , u(x) +v(x)\\) \u306e \\(0 \\leqq x \\leqq 1\\) \u306b\u304a\u3051\u308b\u6700\u5927\u5024\u3092 \\(b _ n , c _ n , d _ n\\) \u3068\u8868\u305b\u3070, (2)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n0 \\leqq d _ n \\leqq b _ n +c _ n\r\n\\]\r\n(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} b _ n = \\displaystyle\\lim _ {n \\rightarrow \\infty} c _ n = 0\r\n\\]\r\n\u306a\u306e\u3067, \u306f\u3055\u307f\u3046\u3061\u306e\u539f\u7406\u3088\u308a\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} d _ n = 0\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} e _ n = \\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( d _ n +r \\right) = \\underline{r}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(n\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3059\u308b\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088. (1)\u3000\\(k\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3059\u308b. \u95a2\u6570 \\((1-x)^n x^k\\) \u306e \\(0 \\leqq x \\leqq 1\\) \u306b\u304a\u3051\u308b\u6700\u5927\u5024\u3092 \\(a _ n\\) … \u7d9a\u304d\u3092\u8aad\u3080