\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\u6b63\u306e\u5b9f\u6570 \\(a\\) \u3068\u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066\u6b21\u306e\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.\r\n\u305f\u3060\u3057, \\(e\\) \u306f\u81ea\u7136\u5bfe\u6570\u306e\u5e95\u3068\u3059\u308b.\r\n\\[\r\ne^a = 1 +a +\\dfrac{a^2}{2 !} +\\cdots +\\dfrac{a^n}{n !} +\\displaystyle\\int _ {0}^{a} \\dfrac{(a-x)^n}{n !} e^x \\, dx\r\n\\]<\/li>\r\n (2)<\/strong>\u3000\u6b63\u306e\u5b9f\u6570 \\(a\\) \u3068\u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066\u6b21\u306e\u4e0d\u7b49\u5f0f\u3092\u793a\u305b.\r\n\\[\r\n\\dfrac{a^{n+1}}{(n+1) !} \\leqq \\displaystyle\\int _ {0}^{a} \\dfrac{(a-x)^n}{n !} e^x \\, dx \\leqq \\dfrac{e^a a^{n+1}}{(n+1) !}\r\n\\]<\/li>\r\n (3)<\/strong>\u3000\u4e0d\u7b49\u5f0f\r\n\\[\r\n\\left| e -\\left( 1 +1 +\\dfrac{1}{2 !} +\\cdots +\\dfrac{1}{n !} \\right) \\right| \\lt 10^{-3}\r\n\\]\r\n\u3092\u6e80\u305f\u3059\u6700\u5c0f\u306e\u6b63\u306e\u6574\u6570 \\(n\\) \u3092\u6c42\u3081\u3088.\r\n\u5fc5\u8981\u306a\u3089\u3070 \\(2 \\lt e \\lt 3\\) \u3067\u3042\u308b\u3053\u3068\u306f\u8a3c\u660e\u306a\u3057\u306b\u7528\u3044\u3066\u3082\u3088\u3044.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u793a\u3057\u305f\u3044\u5f0f\u3092 [A] \u3068\u304a\u304d, \u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n 1*<\/strong>\u3000\\(n = 1\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ {0}^{a} (a-x) e^x \\, dx & = \\left[ (a-x) e^x \\right] _ {0}^{a} +\\displaystyle\\int _ {0}^{a} e^x \\, dx \\\\\r\n& = -a +\\left[ e^x \\right] _ {0}^{a} = e^a -1 -a\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n 2*<\/strong>\u3000\\(n = k\\) \u306e\u3068\u304d\r\n[A] \u304c\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ {0}^{a} \\dfrac{(a-x)^{k+1}}{(k+1) !} e^x \\, dx & = \\left[ \\dfrac{(a-x)^{k+1}}{(k+1) !} e^x \\right] _ {0}^{a} +\\displaystyle\\int _ {0}^{a} \\dfrac{(a-x)^k}{k !} e^x \\, dx \\\\\r\n& = -\\dfrac{a^{k+1}}{(k+1) !} +e^a -\\left( 1 +a +\\cdots +\\dfrac{a^k}{k !} \\right)\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, \\(n = k+1\\) \u306e\u3068\u304d\u3082 [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n \u4ee5\u4e0a\u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n (2)<\/strong><\/p>\r\n \\(0 \\leqq x \\leqq a\\) \u306b\u304a\u3044\u3066, \\(a-x \\geqq 0\\) , \\(1 \\leqq e^x \\leqq e^a\\) \u306a\u306e\u3067\r\n\\[\r\n\\dfrac{(a-x)^n}{n !} \\leqq \\dfrac{(a-x)^n}{n !} e^x \\leqq \\dfrac{(a-x)^n}{n !} e^a\r\n\\]\r\n\\(\\displaystyle\\int _ {0}^{a} (a-x)^n \\, dx = \\dfrac{a^n}{n+1}\\) \u306a\u306e\u3067, \u8fba\u3005\u3092 \\(x\\) \u306b\u3064\u3044\u3066 \\(0 \\rightarrow a\\) \u3067\u7a4d\u5206\u3059\u308c\u3070\r\n\\[\r\n\\dfrac{a^{n+1}}{(n+1) !} \\leqq \\displaystyle\\int _ {0}^{a} \\dfrac{(a-x)^n}{n !} e^x \\, dx \\leqq \\dfrac{e^a a^{n+1}}{(n+1) !}\r\n\\]\r\n (3)<\/strong><\/p>\r\n \\(S_n = e -\\left( 1 +1 +\\dfrac{1}{2 !} +\\cdots +\\dfrac{1}{n !} \\right)\\) \u3068\u304a\u304f.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n
\r\n(1)<\/strong> (2)<\/strong> \u306e\u7d50\u679c\u306b\u3064\u3044\u3066, \\(a=1\\) \u3068\u3059\u308c\u3070\r\n\\[\r\n\\dfrac{1}{(n+1) !} \\leqq S_n \\leqq \\dfrac{e}{(n+1) !}\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070, \\(n = 5\\) \u306e\u3068\u304d\r\n\\[\r\nS_5 \\geqq \\dfrac{1}{6 !} = \\dfrac{1}{720} \\gt 10^{-3}\r\n\\]\r\n\\(n = 6\\) \u306e\u3068\u304d, \\(e \\lt 3\\) \u3082\u7528\u3044\u3066\r\n\\[\r\nS_6 \\leqq \\dfrac{e}{7 !} \\lt \\dfrac{1}{1680} \\lt 10^{-3}\r\n\\]\r\n\\(S_n\\) \u306f \\(n\\) \u306e\u6e1b\u5c11\u95a2\u6570\u3067, \\(S_5 \\gt 10^{-3} \\gt S_6\\) \u3060\u304b\u3089, \u6c42\u3081\u308b \\(n\\) \u306e\u5024\u306f\r\n\\[\r\nn = \\underline{6}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\u6b63\u306e\u5b9f\u6570 \\(a\\) \u3068\u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066\u6b21\u306e\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b. \u305f\u3060\u3057, \\(e\\) \u306f\u81ea\u7136\u5bfe\u6570\u306e\u5e95\u3068\u3059\u308b. \\[ e^a = 1 +a +\\dfrac{a^2}{2 … \u7d9a\u304d\u3092\u8aad\u3080