{"id":10910,"date":"2024-02-04T00:00:49","date_gmt":"2024-02-03T15:00:49","guid":{"rendered":"https:\/\/www.educational-lounge.com\/?p=10910"},"modified":"2024-02-03T18:49:52","modified_gmt":"2024-02-03T09:49:52","slug":"%e6%95%b0%e5%ad%a6%e2%85%b2%e3%81%a7%e7%99%bb%e5%a0%b4%e3%81%99%e3%82%8b%e8%87%aa%e7%84%b6%e5%af%be%e6%95%b0%e3%81%ae%e5%ba%95%e3%80%8c%e3%83%8d%e3%82%a4%e3%83%94%e3%82%a2%e6%95%b0%e3%80%8d%e3%81%ae","status":"publish","type":"post","link":"https:\/\/www.educational-lounge.com\/?p=10910","title":{"rendered":"\u6570\u5b66\u2162\u3067\u767b\u5834\u3059\u308b\u81ea\u7136\u5bfe\u6570\u306e\u5e95\u300c\u30cd\u30a4\u30d4\u30a2\u6570\u300d\u306e\u5b9a\u7fa9\u3068\u51fa\u6240"},"content":{"rendered":"

\u3069\u3046\u3082, \u307f\u306a\u3055\u3093\u3053\u3093\u306b\u3061\u306f\u3002
\n\u9ad8\u6a4b\u4f73\u4f51\u3067\u3059\u3002<\/p>\n

\u4eca\u56de\u306f\u6570\u5b66\u2162\u306b\u767b\u5834\u3059\u308b\u30cd\u30a4\u30d4\u30a2\u6570\\(e\\)\u3068\u3044\u3046\u5b9a\u6570\u306b\u3064\u3044\u3066\u304a\u8a71\u3057\u3057\u307e\u3059\u3002<\/p>\n

\u3053\u306e\u5b9a\u6570\\(e\\)\u306b\u306f\u7d20\u6674\u3089\u3057\u3044\u6027\u8cea\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n

\u95a2\u6570\\(e^x\\)\u306f
\n$$\\frac{d}{dx}e^x=e^x$$
\n$$\\displaystyle\\int e^xdx=e^x+C$$
\n\u3068\u306a\u308a, \u5fae\u5206\u3057\u3066\u3082\u7a4d\u5206\u3057\u3066\u3082\u5f62\u3092\u5909\u3048\u307e\u305b\u3093\u3002\u3055\u3089\u306b, \u3053\u306e\\(e\\)\u3092\u5e95\u3068\u3059\u308b\u5bfe\u6570\\(\\log_ex\\)\u306f, \u81ea\u7136\u5bfe\u6570\u3068\u3088\u3070\u308c, $$\\frac{d}{dx}\\log_e|x|=\\frac{1}{x}$$$$\\int\\frac{1}{x}\\ dx=\\log_e|x|+C$$\u3068\u306a\u308a\u307e\u3059\u3002\u5e95\\(e\\)\u306f\u7701\u7565\u3055\u308c, \\(\\log x\\)\u3068\u8868\u3055\u308c\u308b\u3053\u3068\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n

\u3053\u308c\u3089\u306f\u5fae\u7a4d\u8a08\u7b97\u306b\u304a\u3044\u3066\u5927\u5909\u4fbf\u5229\u306a\u6027\u8cea\u3067\u3059\u3002<\/p>\n

\u3053\u3053\u3067\u767b\u5834\u3059\u308b\u30cd\u30a4\u30d4\u30a2\u6570\\(e\\)\u3068\u306f\u3069\u306e\u3088\u3046\u306a\u6570\u306a\u306e\u3067\u3057\u3087\u3046\u304b\u3002
\n\u5b9a\u7fa9\u3068\u305d\u306e\u51fa\u6240\u3092\u898b\u3066\u884c\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n

\u8a18\u4e8b\u306e\u5f8c\u534a\u306e\u300c\u65e5\u5e38\u306e\u73fe\u8c61\u306b\u304a\u3051\u308b\\(e\\)\u300d\u3067\u306f, \u305c\u3072\u6311\u6226\u3057\u3066\u307b\u3057\u3044\u6975\u9650\u8a08\u7b97\u304c\u3042\u308a\u307e\u3059\u306e\u3067\u30c8\u30e9\u30a4\u3057\u3066\u307f\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n

\u899a\u3048\u3066\u304a\u304d\u305f\u3044\u300c\u30cd\u30a4\u30d4\u30a2\u6570\\(e\\)\u306e\u5b9a\u7fa9\u300d\uff12\u3064<\/h2>\n

\"\"
\n\u30e1\u30b8\u30e3\u30fc\u306a\u5b9a\u7fa9\u306f\u6b21\u306e2\u3064\u3067\u3059\u3002<\/p>\n

\n
\u30cd\u30a4\u30d4\u30a2\u6570\u306e\u30e1\u30b8\u30e3\u30fc\u306a\u5b9a\u7fa9<\/div>\n
\n

$\u2460 \\displaystyle\\lim_{n\\to\\infty}\\left(1+\\frac{1}{n}\\right)^n=e\\\\
\n\u2461 \\lim_{t\\to0}(1+t)^{\\frac{1}{t}}=e$<\/p>\n<\/div>\n<\/div>\n

\u3053\u306e\u4ed6\u306b\u3082, \u30cd\u30a4\u30d4\u30a2\u6570\\(e\\)\u3092\u5b9a\u7fa9\u3059\u308b\u65b9\u6cd5\u304c\u3042\u308a\u307e\u3059\u3002\u4e0a\u306e2\u3064\u306e\u5b9a\u7fa9\u306f\u899a\u3048\u3066\u304a\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n

\\(e\\)\u306e\u8d77\u6e90\u306f1700\u5e74\u4ee3<\/h2>\n

\\(e\\)\u3068\u3044\u3046\u8a18\u53f7\u306f\uff0c1700\u5e74\u4ee3\u306b\u6570\u5b66\u8005\u30aa\u30a4\u30e9\u30fc\u304c\u81ea\u7136\u5bfe\u6570\u306e\u5e95\u3092\u610f\u5473\u3059\u308b\u6570\u3068\u3057\u3066\u7528\u3044\u3066\u5e83\u307e\u308a\u307e\u3057\u305f\u3002<\/p>\n

\n
\n
\u5bfe\u6570\u306e\u6982\u5ff5\u306f1600\u5e74\u5e8f\u76e4\u306b\u30cd\u30a4\u30d4\u30a2, \u30d6\u30ea\u30c3\u30af\u30b9\u3089\u306b\u3088\u3063\u3066\u78ba\u7acb\u3055\u308c, \u5e38\u7528\u5bfe\u6570\u3092\u8868\u3067\u307e\u3068\u3081\u305f\u3082\u306e\u3092\u51fa\u7248\u3057\u3066\u3044\u307e\u3059\u304c, \u5f53\u6642\u306f\u81ea\u7136\u5bfe\u6570\u3092\u8868\u3059\u8a18\u53f7\u306f\u4e00\u822c\u7684\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3067\u3057\u305f\u3002<\/div>\n<\/div>\n<\/div>\n

\u3061\u306a\u307f\u306b, \u30aa\u30a4\u30e9\u30fc\u306f\u3053\u306e\\(e\\)\u4ee5\u5916\u306b\u3082, \u5186\u5468\u7387\\(\\pi\\)\u3084\u4e09\u89d2\u95a2\u6570\\(\\sin\\), \\(\\cos\\), \\(\\tan\\)\u3082\u8a18\u53f7\u5316\u3057, \u73fe\u5728\u4f7f\u308f\u308c\u3066\u3044\u308b\u8a18\u53f7\u3068\u307b\u307c\u540c\u69d8\u306a\u3082\u306e\u3092\u7528\u3044\u3066\u5e83\u3081\u305f\u4eba\u3067\u3082\u3042\u308a\u307e\u3059\u3002<\/p>\n

\u3053\u3053\u3067, \\(e\\)\u306e\u5b9a\u7fa9<\/p>\n

\n
\n

$\u2460 \\displaystyle\\lim_{n\\to\\infty}\\left(1+\\frac{1}{n}\\right)^n=e\\\\
\n\u2461 \\lim_{t\\to0}(1+t)^{\\frac{1}{t}}=e$<\/p>\n<\/div>\n<\/div>\n

\u306b\u3064\u3044\u3066\u8003\u3048\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n

\\(\u2460\\displaystyle\\lim_{n\\to\\infty}\\left(1+\\frac{1}{n}\\right)^n=e\\)\u306b\u3064\u3044\u3066<\/h3>\n

\u3053\u306e\u5b9a\u7fa9\u306e\u7531\u6765\u306f\u9023\u7d9a\u798f\u5229\u3068\u3044\u3046\u8003\u3048\u65b9\u306b\u3042\u308a\u307e\u3059\u3002<\/p>\n

\u8a71\u3092\u7c21\u6f54\u306b\u3057\u307e\u3059\u3002<\/p>\n

\u304a\u91d1\u3092\u3042\u308b\u6a5f\u95a2\u306b\u9810\u3051\u308b\u3053\u3068\u3092\u8003\u3048\u3066\u304f\u3060\u3055\u3044\u3002\u305d\u306e\u6a5f\u95a2\u306f\u4e00\u5b9a\u6642\u9593\u7d4c\u3064\u3068\u9810\u3051\u3066\u3044\u308b\u304a\u91d1\u306b\u5bfe\u3057\u3066\u5229\u76ca\u3092\u3064\u3051\u3066\u304f\u308c\u307e\u3059\u3002<\/p>\n

\u3053\u3053\u3067\u6b21\u306e\u3088\u3046\u306a\u6a5f\u95a2\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n

\n
\n
\u6700\u521d\u306b\u9810\u3051\u305f\u91d1\u984d\u3092,
\n$1$\u5e74\u3067$\\left(1+\\frac{1}{1}\\right)$\u500d\u306b\u3057\u3066\u304f\u308c\u308b\u6a5f\u95a2
\n$\\frac{1}{2}$\u5e74\u3067$\\left(1+\\frac{1}{2}\\right)$\u500d\u306b\u3057\u3066\u304f\u308c\u308b\u6a5f\u95a2
\n$\\frac{1}{3}$\u5e74\u3067$\\left(1+\\frac{1}{3}\\right)$\u500d\u306b\u3057\u3066\u304f\u308c\u308b\u6a5f\u95a2
\n$\\vdots$
\n<\/span>$\\frac{1}{n}$\u5e74\u3067$\\left(1+\\frac{1}{n}\\right)$\u500d\u306b\u3057\u3066\u304f\u308c\u308b\u6a5f\u95a2<\/div>\n<\/div>\n<\/div>\n

\u3053\u306e\u3088\u3046\u306b\u77ed\u3044\u6642\u9593\u3067\u5229\u76ca\u3092\u3064\u3051\u3066\u304f\u308c\u308b\u6a5f\u95a2\u3092\u60f3\u5b9a\u3057, 1\u5e74\u5f8c\u306b\u304a\u91d1\u306f\u3069\u3046\u306a\u3063\u3066\u3044\u308b\u306e\u304b\u8abf\u3079\u3066\u307f\u307e\u3059\u3002<\/p>\n

\\(\\frac{1}{2}\\)\u5e74\u3067\\(\\left(1+\\frac{1}{2}\\right)\\)\u500d\u306b\u3057\u3066\u304f\u308c\u308b\u6a5f\u95a2\u306e\u5834\u5408, 1\u5e74\u30672\u56de\u5229\u76ca\u3092\u3064\u3051\u3066\u304f\u308c\u308b\u306e\u3067, 1\u5e74\u3067\\(\\left(1+\\frac{1}{2}\\right)^2\\)\u500d\u306b\u3057\u3066\u304f\u308c\u308b\u3068\u8003\u3048\u307e\u3059\u3002
\n\\(\\frac{1}{3}\\)\u5e74\u3067\\(\\left(1+\\frac{1}{3}\\right)\\)\u500d\u306b\u3057\u3066\u304f\u308c\u308b\u6a5f\u95a2\u306e\u5834\u5408, 1\u5e74\u30673\u56de\u5229\u76ca\u3092\u3064\u3051\u3066\u304f\u308c\u308b\u306e\u3067, 1\u5e74\u3067\\(\\left(1+\\frac{1}{3}\\right)^3\\)\u500d\u306b\u3057\u3066\u304f\u308c\u308b\u3068\u8003\u3048\u307e\u3059\u3002<\/p>\n

\"\"<\/a><\/p>\n

\u4ee5\u4e0b\u540c\u69d8\u306b\u3057\u3066\u8003\u3048\u3066\u307f\u308b\u3068,<\/p>\n

$1$\u5e74\u3067$\\left(1+\\frac{1}{1}\\right)$\u500d\u306b\u3057\u3066\u304f\u308c\u308b\u3068\u304d, $1+\\frac{1}{1}=2$\u500d
\n$\\frac{1}{2}$\u5e74\u3067$\\left(1+\\frac{1}{2}\\right)$\u500d\u306b\u3057\u3066\u304f\u308c\u308b\u3068\u304d, $\\left(1+\\frac{1}{2}\\right)^2=2.25$\u500d
\n$\\frac{1}{3}$\u5e74\u3067$\\left(1+\\frac{1}{3}\\right)$\u500d\u306b\u3057\u3066\u304f\u308c\u308b\u3068\u304d, $\\left(1+\\frac{1}{3}\\right)^3=2.370\\cdots\\cdots$\u500d
\n$\\vdots$
\n$\\frac{1}{100}$\u5e74\u3067$\\left(1+\\frac{1}{100}\\right)$\u500d\u306b\u3057\u3066\u304f\u308c\u308b\u3068\u304d, $\\left(1+\\frac{1}{100}\\right)^{100}=2.704\\cdots\\cdots $\u500d
\n$\\vdots$
\n$\\frac{1}{1000}$\u5e74\u3067$\\left(1+\\frac{1}{1000}\\right)$\u500d\u306b\u3057\u3066\u304f\u308c\u308b\u3068\u304d,$ \\left(1+\\frac{1}{1000}\\right)^{1000}=2.716\\cdots\\cdots$\u500d
\n$\\vdots$
\n$\\frac{1}{10000}$\u5e74\u3067$\\left(1+\\frac{1}{10000}\\right)$\u500d\u306b\u3057\u3066\u304f\u308c\u308b\u3068\u304d,$\\left(1+\\frac{1}{10000}\\right)^{10000}=2.718\\cdots\\cdots$\u500d
\n$\\vdots$<\/p>\n

\u3053\u308c\u3089\u306e\u7d50\u679c\u304b\u3089, \\(2.71\\cdots\\cdots\\)\u3068\u3044\u3046\u5024\u306b\u8fd1\u3065\u3044\u3066\u3044\u304d\u305d\u3046\u3068\u3044\u3046\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002\u5b9f\u306f, \u6570\u5217\\(\\left\\{\\left(1+\\frac{1}{n}\\right)^n\\right\\}\\)\u306f\\(2.71828\\cdots\\cdots\\)\u3068\u3044\u3046\u5024\u306b\u53ce\u675f\u3059\u308b\u3053\u3068\u304c\u5206\u304b\u3063\u3066\u304a\u308a, \u3053\u308c\u3092$$\\displaystyle\\lim_{n\\to\\infty}\\left(1+\\frac{1}{n}\\right)^n=e$$
\n\u3068\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n\n

\\(\u2461\\displaystyle\\lim_{t\\to0}(1+t)^{\\frac{1}{t}}=e\\)\u306b\u3064\u3044\u3066<\/h3>\n

\u5c0e\u95a2\u6570\u306e\u5b9a\u7fa9\u306b\u3057\u305f\u304c\u3063\u3066\\(y=\\log_ax\\)\u3068\u3044\u3046\u95a2\u6570\u3092\u5fae\u5206\u3059\u308b\u3053\u3068\u3092\u8003\u3048\u3066\u307f\u307e\u3057\u3087\u3046\u3002
\n\\begin{align*}
\n&\\displaystyle\\lim_{h\\to0}\\cfrac{\\log_a(x+h)-\\log_ax}{h}\\\\
\n=&\\lim_{h\\to0}\\cfrac{\\log_a\\left(\\frac{x+h}{x}\\right)}{h}\\\\
\n=&\\lim_{h\\to0}\\cfrac{\\log_a\\left(1+\\frac{h}{x}\\right)}{h}\\\\
\n=&\\lim_{h\\to0}\\cfrac{1}{h}\\log_a\\left(1+\\frac{h}{x}\\right)\\\\
\n\\end{align*}
\n\u3053\u3053\u3067, \\(\\frac{h}{x}=t\\)\u3068\u304a\u304f\u3068, \\(h\\to0\\)\u306e\u3068\u304d\\(t\\to0\\)\u3067, \\(h=xt\\)\u3068\u306a\u308b\u304b\u3089,
\n\\begin{align*}
\n&\\displaystyle\\lim_{h\\to0}\\cfrac{\\log_a(x+h)-\\log_ax}{h}\\\\
\n=&\\lim_{h\\to0}\\cfrac{1}{h}\\log_a\\left(1+\\frac{h}{x}\\right)\\\\
\n=&\\lim_{t\\to0}\\cfrac{1}{tx}\\log_a(1+t)\\\\
\n=&\\lim_{t\\to0}\\cfrac{1}{x}\\cdot\\cfrac{1}{t}\\log_a(1+t)\\\\
\n=&\\lim_{t\\to0}\\cfrac{1}{x}\\log_a(1+t)^{\\frac{1}{t}}\\\\
\n\\end{align*}
\n\u3053\u306e\u3068\u304d, \\(\\displaystyle\\lim_{t\\to0}(1+t)^{\\frac{1}{t}}\\)\u306e\u6975\u9650\u3092\u8abf\u3079\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u4e0a\u3068\u540c\u69d8\u306e\u5f0f\u306b\u306a\u308a\u307e\u3059\u304c,
\n$$t=\\cfrac{1}{10}\u306e\u3068\u304d, \\left(1+\\cfrac{1}{10}\\right)^{10}=2.593\\cdots\\cdots$$
\n$$\\vdots$$
\n$$t=\\cfrac{1}{100}\u306e\u3068\u304d, \\left(1+\\cfrac{1}{100}\\right)^{100}=2.704\\cdots\\cdots$$
\n$$\\vdots$$
\n$$t=\\cfrac{1}{1000}\u306e\u3068\u304d, \\left(1+\\cfrac{1}{1000}\\right)^{1000}=2.716\\cdots\\cdots$$
\n$$\\vdots$$
\n$$t=\\cfrac{1}{10000}\u306e\u3068\u304d, \\left(1+\\cfrac{1}{10000}\\right)^{10000}=2.718\\cdots\\cdots$$
\n$$\\vdots$$
\n\u3068\u306a\u308a, \\(2.71828\\cdots\\cdots\\)\u3068\u3044\u3046\u5024\u306b\u53ce\u675f\u3059\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066,
\n$$\\lim_{t\\to0}(1+t)^{\\frac{1}{t}}=e$$
\n\u3068\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n

\u3055\u3089\u306b, \u66f2\u7dda\\(y=a^x\\)\u306e\u70b9\\((0, 1)\\)\u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u50be\u304d\u304c1\u306b\u306a\u308b\\(a\\)\u306e\u5024\u3092\u6c42\u3081\u3066\u307f\u307e\u3057\u3087\u3046\u3002\u3064\u307e\u308a,$$\\displaystyle\\lim_{x\\to0}\\cfrac{a^x-1}{x}=1$$\u3068\u306a\u308b\\(a\\)\u3092\u6c42\u3081\u307e\u3059\u3002<\/p>\n

\"\"<\/a><\/p>\n

\\(a^x-1=t\\)\u3068\u304a\u304f\u3068, \\(x\\to0\\)\u306e\u3068\u304d, \\(t\\to0\\)\u3067\u3042\u308a, \\(a^x=t+1\\)\u304b\u3089, \\(x=\\log_a(t+1)\\)\u306a\u306e\u3067, $$\\lim_{x\\to0}\\cfrac{a^x-1}{x}=\\lim_{t\\to0}\\cfrac{t}{\\log_a(t+1)}=\\lim_{t\\to0}\\cfrac{1}{\\frac{1}{t}\\log_a(1+t)}=\\lim_{t\\to0}\\cfrac{1}{\\frac{1}{t}\\log_a(1+t)^{\\frac{1}{t}}}=\\cfrac{1}{\\log_ae}$$
\n\u3068\u306a\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066, \\(\\log_ae=1\\)\u304b\u3089\\(a=e\\)\u3068\u306a\u308b\u308f\u3051\u3067\u3059\u3002\u3088\u3063\u3066, $$\\displaystyle\\lim_{x\\to0}\\cfrac{e^x-1}{x}=1$$\u3068\u306a\u308a\u307e\u3059\u3002\u5b9f\u306f\u3053\u308c\u3092\\(e\\)\u306e\u5b9a\u7fa9\u3068\u3059\u308b\u3053\u3068\u3082\u3042\u308a\u307e\u3059\u3002<\/p>\n

\u3053\u306e\u3088\u3046\u306b\\(e\\)\u306f\u3044\u308d\u3044\u308d\u306a\u3068\u3053\u308d\u304b\u3089\u51fa\u73fe\u3059\u308b\u306e\u3067\u3059\u3002\u3069\u308c\u304b1\u3064\u3092\u5b9a\u7fa9\u3068\u3059\u308c\u3070, \u4ed6\u306e\u5f0f\u306f\u6027\u8cea\u3068\u3057\u3066\u5c0e\u3051\u307e\u3059\u3002<\/p>\n\n

\\(e\\)\u306f\u65e5\u5e38\u306e\u73fe\u8c61\u306b\u3082\u6f5c\u3093\u3067\u3044\u308b<\/h2>\n

\"\u6570\u5b66\u306e\u5206\u539a\u3044\u554f\u984c\u96c6\u306f\u300c\u554f\u984c\u8f9e\u66f8\u300d\u3068\u3057\u3066\u5229\u7528\u3057\u3088\u3046\"
\n\u65e5\u5e38\u306e\u4e2d\u306b\u3082\u81ea\u7136\u5bfe\u6570\u306e\u5e95\\(e\\)\u304c\u73fe\u308c\u308b\u3053\u3068\u304c\u3042\u308a\u307e\u3059\u3002\u4eca\u56de\u306f2\u3064\u7d39\u4ecb\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n

\u30d7\u30ec\u30bc\u30f3\u30c8\u4ea4\u63db\u4f1a<\/h3>\n

\\(n\\)\u4eba\u3067\u30d7\u30ec\u30bc\u30f3\u30c8\u4ea4\u63db\u4f1a\u3092\u958b\u304f\u3068\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n

\u5404\u81ea\u30d7\u30ec\u30bc\u30f3\u30c8\u30921\u3064\u7528\u610f\u3057, \u81ea\u5206\u304c\u8ab0\u306e\u30d7\u30ec\u30bc\u30f3\u30c8\u3092\u53d7\u3051\u53d6\u308b\u304b\u3092\u304f\u3058\u5f15\u304d\u3067\u6c7a\u3081\u308b\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d, \u5168\u54e1\u304c\u81ea\u5206\u306e\u7528\u610f\u3057\u305f\u30d7\u30ec\u30bc\u30f3\u30c8\u306b\u5f53\u305f\u3089\u306a\u3044\u3068\u304d\u6210\u529f\u3068\u3057\u307e\u3059\u3002<\/p>\n

1\u56de\u306e\u304f\u3058\u5f15\u304d\u3067\u6210\u529f\u3059\u308b\u78ba\u7387\u306f\u3069\u3046\u306a\u308b\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n

\"\"<\/a><\/p>\n

\u4e0a\u306e\u3088\u3046\u306a\u30d7\u30ec\u30bc\u30f3\u30c8\u306e\u53d7\u3051\u53d6\u308a\u65b9\u3092\u5b8c\u5168\u9806\u5217\u3084\u652a\u4e71\u9806\u5217\u3068\u547c\u3093\u3060\u308a\u3057\u307e\u3059\u3002<\/p>\n

\u305d\u3057\u3066, \\(n\\)\u3092\u5927\u304d\u304f\u3057\u3066\u3044\u304f\u3068\u6210\u529f\u3059\u308b\u78ba\u7387\u306f\\(\\frac{1}{e}=0.367\\cdots\\cdots\\)\u306b\u8fd1\u3065\u3044\u3066\u3044\u304d\u307e\u3059\u3002\u5b9f\u969b\u306b\u3053\u308c\u3092\u8a08\u7b97\u3059\u308b\u3068\u304d\u306f, \u4eca\u56de\u7d39\u4ecb\u3057\u305f\u5b9a\u7fa9\u4ee5\u5916\u306e\u3053\u3068\u3092\u8003\u3048\u308b\u306e\u3067\u3053\u3053\u3067\u306f\u7701\u7565\u3057\u307e\u3059\u3002<\/p>\n

\u300c\u3010\u30ec\u30d9\u30eb\u5225\u3011\u6570\u5b66\u2162\u306e\u5fae\u7a4d\u5206\u3092\u653b\u7565\u3059\u308b\u52c9\u5f37\u6cd5\u300d<\/a>\u3067\u7d39\u4ecb\u3057\u305f\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u306b\u95a2\u4fc2\u3057\u307e\u3059\u3002
\n

\"\u3010\u30ec\u30d9\u30eb\u5225\u3011\u6570\u5b66\u2162\u306e\u5fae\u7a4d\u5206\u3092\u653b\u7565\u3059\u308b\u52c9\u5f37\u6cd5\"\u6570\u5b66<\/span><\/div>
\u3010\u30ec\u30d9\u30eb\u5225\u3011\u6570\u5b66\u2162\u306e\u5fae\u7a4d\u5206\u3092\u653b\u7565\u3059\u308b\u52c9\u5f37\u6cd5<\/div>
\u3069\u3046\u3082\u3001\u307f\u306a\u3055\u3093\u3053\u3093\u306b\u3061\u306f\u3002\u9ad8\u6a4b\u4f73\u4f51\u3067\u3059\u3002 \u4eca\u56de\u306f\u6570\u5b66\u2162\u306e\u5358\u5143\u300c\u5fae\u5206\u7a4d\u5206\u300d\u306e\u5b66\u7fd2\u306e\u30dd\u30a4\u30f3\u30c8\u306b\u95a2\u3057\u3066\u304a\u8a71\u3057\u3057\u307e\u3059\u3002 \u5fae\u5206\u7a4d\u5206\u306e\u521d\u5b66\u8005\u306f\u8a08\u7b97\u3092\u4ed5\u4e0a\u3052\u308b\u3068\u3053\u308d\u304b\u3089 \u307e\u305a\u306f\u8a08\u7b97\u3092\u3057\u3063\u304b\u308a\u4ed5\u4e0a\u3052\u307e\u3057\u3087\u3046\u3002 \u5fae\u5206\u7de8 \u6570\u5b66\u2161\u3067\u306f\u591a\u9805\u5f0f\u3067\u8868\u3055\u308c\u308b\u95a2\u6570\u3092\u6271\u3044\u307e\u3057\u305f\u304c\u3001\u6570\u5b66\u2162\u3067\u306f\u300c\u6975\u9650\u300d\u306e\u5358\u5143\u3067\u3044\u308d\u3044\u308d\u306a\u95a2\u6570\u306e\u6975\u9650\u306e...<\/div><\/a><\/div><\/p>\n

\u307e\u305f, \u3053\u306e\u8a71\u306f\u5e2d\u66ff\u3048\u306e\u304a\u8a71\u3068\u8003\u3048\u308b\u3053\u3068\u3082\u3067\u304d\u3066, \u305d\u306e\u5834\u5408\u300c\u5e2d\u66ff\u3048\u3092\u3057\u305f\u6642\u306b\u5168\u54e1\u304c\u5225\u306e\u5e2d\u306b\u306a\u308b\u78ba\u7387\u304c\\(\\frac{1}{e}\\)\u300d\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u7a00\u306a\u73fe\u8c61\u304c\u8d77\u3053\u308b\u78ba\u7387<\/h3>\n

\u73fe\u884c\u904e\u7a0b\u306b\u304a\u3044\u3066\u6570\u5b66B\u300c\u78ba\u7387\u5206\u5e03\u3068\u7d71\u8a08\u7684\u306a\u63a8\u6e2c\u300d\u3067\u767b\u5834\u3059\u308b\u4e8c\u9805\u5206\u5e03\u306e\u7279\u6b8a\u306a\u5834\u5408\u3092\u8003\u3048\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n

\u65b0\u8ab2\u7a0b\u306b\u306a\u308a\u3053\u306e\u5206\u91ce\u304c\u5fc5\u4fee\u306b\u306a\u308b\u3053\u3068\u304b\u3089, \u3053\u306e\u8a71\u3092\u984c\u6750\u306b\u3057\u305f\u554f\u984c\u304c\u5165\u8a66\u3067\u51fa\u984c\u3055\u308c\u308b\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3002
\n\u73fe\u884c\u904e\u7a0b\u3067\u6570\u5b66B\u300c\u78ba\u7387\u5206\u5e03\u3068\u7d71\u8a08\u7684\u306a\u63a8\u6e2c\u300d\u3092\u9078\u629e\u3057\u306a\u3044\u4eba\u306f, \u6975\u9650\u306e\u8a08\u7b97\u3060\u3051\u3067\u3082\u898b\u3066\u307f\u3066\u4e0b\u3055\u3044\u3002<\/p>\n

\u78ba\u7387\u5909\u6570\\(X\\)\u304c\u4e8c\u9805\u5206\u5e03\\(B(n, p)\\)\u306b\u5f93\u3046\u3068\u304d,\\(X=k\\)\u3068\u306a\u308b\u78ba\u7387\u306f$$P(X=k)=_n{\\rm C}_kp^k(1-p)^{n-k}$$\u3068\u306a\u308a\u307e\u3059\u3002\u671f\u5f85\u5024\\(E(X)=np\\)\u3092\u4e00\u5b9a\u306e\u5024\\(\\lambda\\)\u3059\u306a\u308f\u3061\\(np=\\lambda\\)\u3068\u3057, \\(n\\to\\infty\\)\u3068\u3057\u305f\u6975\u9650\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n

\u73fe\u884c\u904e\u7a0b\u3067\u6570\u5b66B\u300c\u78ba\u7387\u5206\u5e03\u3068\u7d71\u8a08\u7684\u306a\u63a8\u6e2c\u300d\u3092\u9078\u629e\u3057\u306a\u3044\u4eba\u306f, \u5206\u5e03\u306a\u3069\u306e\u7d71\u8a08\u7684\u306a\u8a71\u306f\u7f6e\u3044\u3066\u304a\u3044\u3066, \u3053\u306e\u6975\u9650\u306e\u8a08\u7b97\u3092\u8003\u3048\u3066\u307f\u307e\u3057\u3087\u3046\uff01
\n<\/p>\n

\\(\\lambda=1\\)\u3068\u3057\u305f\u6975\u9650\u306e\u51fa\u984c\u4f8b<\/span><\/h4>\n

\\(\\lambda=1\\)\u3068\u3057\u305f\u6975\u9650\u306f\u6614\u4e5d\u5dde\u5927\u5b66\u3067\u51fa\u984c\u3055\u308c\u3066\u3044\u307e\u3057\u305f\u3002<\/p>\n

\n
\u4f8b\u984c<\/div>\n
\n

$np=\\lambda$\u306e\u3068\u304d, $\\displaystyle\\lim_{n\\to\\infty}{}_n{\\rm C}_kp^k(1-p)^{n-k}$\u3092\u6c42\u3081\u3088\u3002\u305f\u3060\u3057, ${}_n{\\rm C}_k$\u306f\u76f8\u7570\u306a\u308b$n$\u500b\u306e\u3082\u306e\u304b\u3089$k$\u500b\u306e\u3082\u306e\u3092\u9078\u3076\u7d44\u307f\u5408\u308f\u305b\u306e\u7dcf\u6570\u3068\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n

\\begin{align*}
\n&\\displaystyle\\lim_{n\\to\\infty}{}_n{\\rm C}_kp^k(1-p)^{n-k}\\\\
\n=&\\lim_{n\\to\\infty}\\cfrac{n!}{k!(n-k)!}\\left(\\cfrac{\\lambda}{n}\\right)^k\\left(1-\\cfrac{\\lambda}{n}\\right)^{n-k}\\\\
\n=&\\lim_{n\\to\\infty}\\cfrac{\\lambda^k}{k!}\\cdot\\cfrac{n(n-1)(n-2)\\cdots\\cdots(n-k+1)}{n^k}\\cdot\\left(1-\\cfrac{\\lambda}{n}\\right)^{n-k}\\\\
\n=&\\lim_{n\\to\\infty}\\cfrac{\\lambda^k}{k!}\\cdot\\cfrac{n}{n}\\cdot\\cfrac{n-1}{n}\\cdot\\cfrac{n-2}{n}\\cdot\\cdots\\cdots\\cdot\\cfrac{n-(k-1)}{n}\\cdot\\left(1-\\cfrac{\\lambda}{n}\\right)^{n-k}\\\\
\n=&\\lim_{n\\to\\infty}\\cfrac{\\lambda^k}{k!}\\cdot\\left(1-\\cfrac{1}{n}\\right)\\left(1-\\cfrac{2}{n}\\right)\\cdots\\cdots\\left(1-\\cfrac{k-1}{n}\\right)\\cdot\\left(1-\\cfrac{\\lambda}{n}\\right)^{n-k}\\\\
\n\\end{align*}<\/p>\n

\u3053\u3053\u3067,
\n$$\\lim_{n\\to\\infty}\\left(1-\\cfrac{1}{n}\\right)\\left(1-\\cfrac{2}{n}\\right)\\cdots\\cdots\\left(1-\\cfrac{k-1}{n}\\right)=1$$
\n\u3067\u3042\u308a, \u3055\u3089\u306b\\(\\displaystyle\\lim_{n\\to\\infty}\\left(1-\\frac{1}{n}\\right)^n=\\lim_{n\\to\\infty}\\left(1-\\frac{1}{n}\\right)^{-n\\cdot(-1)}=e^{-1}\\)\u3068\u306a\u308b\u3053\u3068\u3092\u8a8d\u3081\u3066,
\n$$\\lim_{n\\to\\infty}\\left(1-\\cfrac{\\lambda}{n}\\right)^{n-k}=\\lim_{n\\to\\infty}\\left(1-\\cfrac{\\lambda}{n}\\right)^{-\\frac{n}{\\lambda}\\cdot\\left(-\\frac{\\lambda}{n}\\right)(n-k)}=\\lim_{n\\to\\infty}\\left(1-\\cfrac{\\lambda}{n}\\right)^{-\\frac{n}{\\lambda}\\cdot\\left(1-\\frac{k}{n}\\right)(-\\lambda)}=e^{-\\lambda}$$
\n\u3068\u306a\u308b\u304b\u3089,
\n$$\\lim_{n\\to\\infty}{}_n{\\rm C}_kp^k(1-p)^{n-k}=\\cfrac{\\lambda^ke^{-\\lambda}}{k!}$$
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u3053\u308c\u306f\u7a00\u306b\u3057\u304b\u8d77\u3053\u3089\u306a\u3044\u3053\u3068\u304c\u5358\u4f4d\u6642\u9593\u3042\u305f\u308a\u306b\u8d77\u3053\u308b\u78ba\u7387\u3092\u8868\u3057\u3066\u304a\u308a, \u78ba\u7387\u5206\u5e03\u306f\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u3068\u547c\u3070\u308c\u307e\u3059\u3002<\/p>\n\n

\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306e\u5b9f\u8df5<\/span><\/h4>\n

\u4f8b\u3048\u3070, \u79c1\u304cEducational Lounge\u306b\u8a18\u4e8b\u306e\u57f7\u7b46\u3092\u3059\u308b\u3068\u304d, 1\u672c\u306e\u539f\u7a3f\u306e\u4e2d\u306b\u73fe\u308c\u308b\u8aa4\u5b57\u6570\u306e\u5e73\u5747\u304c1\u6587\u5b57\u3060\u3068\u3057\u307e\u3059\u3002\u8a18\u4e8b1\u672c\u306e\u4e2d\u306e\u8aa4\u5b57\u6570\\(X\\)\u306f\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306b\u5f93\u3046\u3068\u307f\u306a\u305b, \u4e0a\u306e\u4f8b\u3060\u3068\\(\\lambda=1\\)\u306e\u5834\u5408\u3067\u3059\u3002\\(k\\)\u6587\u5b57\u306e\u8aa4\u5b57\u304c\u3042\u308b\u78ba\u7387\u306f$$P(X=k)=\\cfrac{e^{-1}}{k!}$$\u3068\u306a\u308a\u307e\u3059\u3002\u5b9f\u969b\u6b21\u306e\u8a18\u4e8b\u3092\u57f7\u7b46\u3057\u305f\u3068\u304d\u306b1\u6587\u5b57\u8aa4\u5b57\u304c\u3042\u308b\u78ba\u7387\u306f$$P(X=1)=\\cfrac{e^{-1}}{1!}=\\cfrac{1}{e}=0.367\\cdots\\cdots$$
\n\u3068\u306a\u308a\u307e\u3059\u3002\u3082\u3061\u308d\u3093\u57f7\u7b46\u3057\u7d42\u308f\u3063\u305f\u5f8c\u306b\u30c1\u30a7\u30c3\u30af\u3057\u307e\u3059\u304b\u3089, \u516c\u958b\u3059\u308b\u307e\u3067\u306b\u306f\u4fee\u6b63\u3055\u308c\u3066\u3044\u307e\u3059\u304c\u3002<\/p>\n

\u3053\u306e\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u306f\u65e5\u5e38\u3067\u3082\u3088\u304f\u4f7f\u308f\u308c\u308b\u78ba\u7387\u5206\u5e03\u3067, \u4f8b\u3048\u3070\u3042\u308b\u5730\u57df\u30671\u65e5\u306b\u8d77\u304d\u308b\u4ea4\u901a\u4e8b\u6545\u306e\u4ef6\u6570\u306a\u3069\u306b\u3082\u5229\u7528\u3055\u308c\u307e\u3059\u3002<\/p>\n

\u30dd\u30a2\u30bd\u30f3\u5206\u5e03\u4ee5\u5916\u306b\u3082\u6b63\u898f\u5206\u5e03\u306a\u3069\u306e\u6709\u540d\u306a\u78ba\u7387\u5206\u5e03\u306b\u304a\u3044\u3066\u81ea\u7136\u5bfe\u6570\u306e\u5e95\\(e\\)\u306f\u3088\u304f\u767b\u5834\u3057\u307e\u3059\u3002<\/p>\n

\u304a\u308f\u308a\u306b<\/h2>\n

\"\"
\n\u4eca\u56de\u306f\u6570\u5b66\u2162\u306b\u767b\u5834\u3059\u308b\u30cd\u30a4\u30d4\u30a2\u6570\\(e\\)\u3068\u3044\u3046\u5b9a\u6570\u306b\u3064\u3044\u3066\u304a\u8a71\u3057\u3057\u307e\u3057\u305f\u3002<\/p>\n

\u69d8\u3005\u306a\u3068\u3053\u308d\u304b\u3089\\(e=2.71828\\cdots\\cdots\\)\u3068\u3044\u3046\u5b9a\u6570\u304c\u51fa\u73fe\u3059\u308b\u306e\u3067, \u5b9a\u7fa9\u306e\u4ed5\u65b9\u306f\u3044\u308d\u3044\u308d\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u3046\u3061\u3069\u308c\u304b\u3092\u5b9a\u7fa9\u3068\u3059\u308c\u3070\u4ed6\u306f\u6027\u8cea\u3068\u3057\u3066\u5c0e\u3051\u307e\u3059\u3002<\/p>\n

\u3055\u3089\u306b, \u81ea\u7136\u5bfe\u6570\u306e\u5e95\\(e\\)\u306f\u65e5\u5e38\u306b\u3082\u96a0\u308c\u3066\u3044\u308b\u3053\u3068\u304c\u4f1d\u308f\u308b\u3068\u5b09\u3057\u3044\u3067\u3059\u3002\u5b9f\u969b, \u305d\u308c\u3092\u5b9f\u611f\u3059\u308b\u3053\u3068\u306f\u304a\u305d\u3089\u304f\u306a\u3044\u3068\u601d\u3044\u307e\u3059\u304c, \u975e\u5e38\u306b\u91cd\u8981\u306a\u6570\u3067\u3059\u3002<\/p>\n

\u305d\u308c\u3067\u306f,\u4eca\u56de\u306f\u3053\u306e\u8fba\u3067\uff01
\n