\\(e\\) \u3092\u81ea\u7136\u5bfe\u6570\u306e\u5e95, \u3059\u306a\u308f\u3061 \\(e = \\displaystyle\\lim _ {t \\rightarrow \\infty} \\left( 1 +\\dfrac{1}{t} \\right)^t\\) \u3068\u3059\u308b.\r\n\u3059\u3079\u3066\u306e\u6b63\u306e\u5b9f\u6570 \\(x\\) \u306b\u5bfe\u3057, \u6b21\u306e\u4e0d\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.\r\n\\[\r\n\\left( 1 +\\dfrac{1}{x} \\right)^x \\lt e \\lt \\left( 1 +\\dfrac{1}{x} \\right)^{x +\\frac{1}{2}}\r\n\\]\r\n
\\(s = 1 +\\dfrac{1}{x}\\) \u3068\u304a\u304f\u3068, \\(x \\gt 0\\) \u3088\u308a \\(s \\gt 1\\) .
\r\n\u307e\u305f\r\n\\[\r\nx = \\dfrac{1}{s-1} , \\quad x +\\dfrac{1}{2} = \\dfrac{s+1}{2 (s-1)} \\quad ... [1]\r\n\\]\r\n\u793a\u3057\u305f\u3044\u4e0d\u7b49\u5f0f\u306e\u8fba\u3005\u306b\u3064\u3044\u3066, \u81ea\u7136\u5bfe\u6570\u3092\u3068\u308a, [1] \u3092\u7528\u3044\u3066\u5909\u5f62\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nx \\log \\left( 1 +\\dfrac{1}{x} \\right) & \\lt 1 \\lt \\left( x +\\dfrac{1}{2} \\right) \\log \\left( 1 +\\dfrac{1}{x} \\right) \\\\\r\n\\dfrac{1}{s-1} \\log s & \\lt 1 \\lt \\dfrac{s+1}{2 (s-1)} \\log s\r\n\\end{align}\\]\r\n\u3055\u3089\u306b, \\(s-1 \\gt 0\\) \u306a\u306e\u3067\r\n\\[\r\n\\log s \\lt s-1 \\lt \\dfrac{s+1}{2} \\log s \\quad ... \\text{[\uff0a]}\r\n\\]\r\n\u3086\u3048\u306b, \u4e0d\u7b49\u5f0f [\uff0a] \u3092\u793a\u305b\u3070\u3088\u3044.<\/p>\r\n\r\n
\\(f(s) = s -1 -\\log s\\) \u3068\u304a\u3051\u3070\r\n\\[\r\nf'(s) = 1 -\\dfrac{1}{s} \\gt 0 \\quad ( \\ \\text{\u2235} \\ s \\gt 1 \\ )\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(s)\\) \u306f\u5358\u8abf\u5897\u52a0\u3067\u3042\u308a\r\n\\[\\begin{align}\r\nf(s) & \\gt f(1) = 0 \\\\\r\n\\text{\u2234} \\quad \\log s & \\lt s-1 \\quad ... [2]\r\n\\end{align}\\]\r\n\\(g(s) = \\dfrac{s+1}{2} \\log s -(s-1)\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{align}\r\ng'(s) & = \\dfrac{1}{2} \\log s +\\dfrac{s+1}{2} \\cdot \\dfrac{1}{s} -1 \\\\\r\n& = \\dfrac{1}{2} \\log s -\\dfrac{1}{2} +\\dfrac{1}{2s}\r\n\\end{align}\\]\r\n\u3055\u3089\u306b\r\n\\[\\begin{align}\r\ng''(s) & = \\dfrac{1}{2} \\left( \\dfrac{1}{s} -\\dfrac{1}{s^2} \\right) \\\\\r\n& = \\dfrac{s-1}{2s^2} \\gt 0 \\quad \\quad ( \\ \\text{\u2235} \\ s-1 \\gt 0 \\ )\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(g'(s)\\) \u306f\u5358\u8abf\u5897\u52a0\u3067\u3042\u308a\r\n\\[\r\ng'(s) \\gt g'(1) = 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(g(s)\\) \u306f\u5358\u8abf\u5897\u52a0\u3067\u3042\u308a\r\n\\[\\begin{align}\r\ng(s) & \\gt g(1) = 0 \\\\\r\n\\text{\u2234} \\quad s-1 & \\lt \\dfrac{s+1}{s} \\log s \\quad ... [3]\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, [2] [3] \u3088\u308a, [\uff0a] \u304c\u6210\u7acb\u3059\u308b\u306e\u3067, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(e\\) \u3092\u81ea\u7136\u5bfe\u6570\u306e\u5e95, \u3059\u306a\u308f\u3061 \\(e = \\displaystyle\\lim _ {t \\rightarrow \\infty} \\left( 1 +\\dfrac{1}{t} \\right)^t\\) \u3068\u3059\u308b. \u3059 … \u7d9a\u304d\u3092\u8aad\u3080