{"id":56597,"date":"2014-09-15T04:08:03","date_gmt":"2014-09-14T20:08:03","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=56597"},"modified":"2021-10-06T16:15:15","modified_gmt":"2021-10-06T08:15:15","slug":"%e5%8f%a6%e4%b8%80%e5%80%8b%e9%87%8d%e8%a6%81%e7%9a%84%e7%84%a1%e7%90%86%e6%95%b8%ef%bc%9ae-%ef%bc%88another-important-irrational-number%ef%bc%9ae%ef%bc%89","status":"publish","type":"post","link":"http:\/\/localhost\/%e5%8f%a6%e4%b8%80%e5%80%8b%e9%87%8d%e8%a6%81%e7%9a%84%e7%84%a1%e7%90%86%e6%95%b8%ef%bc%9ae-%ef%bc%88another-important-irrational-number%ef%bc%9ae%ef%bc%89\/","title":{"rendered":"\u53e6\u4e00\u500b\u91cd\u8981\u7684\u7121\u7406\u6578\uff1ae \uff08Another important irrational number\uff1ae\uff09"},"content":{"rendered":"

\u53e6\u4e00\u500b\u91cd\u8981\u7684\u7121\u7406\u6578\uff1ae \uff08Another important irrational number\uff1ae\uff09<\/strong><\/span>
\n\u81fa\u5317\u5e02\u7acb\u548c\u5e73\u9ad8\u4e2d\u6559\u5e2b\u9ec3\u4fca\u744b<\/strong><\/span><\/p>\n

\u73fe\u4eca\u9ad8\u4e2d\u4e00\u5e74\u7d1a\u8ab2\u7a0b\u4e2d\u7684\u3008\u6578\u8207\u5f0f\u3009\u55ae\u5143\u88e1\uff0c\u7c21\u55ae\u5730\u8a0e\u8ad6\u4e86\u6578\u7cfb\u5bb6\u65cf\u4e2d\u7684\u5404\u500b\u6210\u54e1\u3002\u5176\u4e2d\uff0c\u7121\u7406\u6578\u6700\u4ee4\u4e00\u822c\u4eba\u611f\u5230\u964c\u751f\u3001\u7121\u6cd5\u6349\u6478\u3002\u6559\u6750\u4e2d\u9664\u4e86\u4ecb\u7d39\u8af8\u5982 \\(\\sqrt{n}\\)\u00a0\u4ee5\u53ca \\(a+\\sqrt{b}\\)\u00a0<\/i>\u985e\u7684\u5e38\u898b\u7121\u7406\u6578\u5916\uff0c\u4e5f\u4ecb\u7d39\u4e86\u5927\u5bb6\u719f\u77e5\u7684\u7121\u7406\u6578\uff0d\u5713\u5468\u7387 \\(\\pi\\)\u3002<\/p>\n

\u7136\u800c\uff0c\u6211\u5011\u4e5f\u77e5\u9053\uff0c\u5be6\u6578\u7dda\u4e0a\u5bc6\u5bc6\u9ebb\u9ebb\u5730\u4f48\u6eff\u4e86\u7121\u7aae\u591a\u500b\u7121\u7406\u6578\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u6d69\u701a\u7684\u5be6\u6578\u4e16\u754c\u88e1\uff0c\u9664\u4e86\u4e0a\u8ff0\u5e38\u898b\u7121\u7406\u6578\u4e4b\u5916\uff0c\u60f3\u5fc5\u5c1a\u6709\u5176\u5b83\u5fdd\u70ba\u4eba\u77e5\u7684\u6210\u54e1\u3002\u9664\u4e86 \\(\\pi\\)\u00a0<\/i>\u4e4b\u5916\uff0c\u53e6\u4e00\u500b\u8457\u540d\u7684\u6210\u54e1\u70ba\u81ea\u7136\u5c0d\u6578\u7684\u5e95\u6578 \\(e\\)\u3002\u81f3\u65bc \\(e\\) \u662f\u4ec0\u9ebc\u6771\u6771\u5462\uff1f\u4ee5\u4e0b\u6211\u5011\u8aaa\u5206\u660e\u3002<\/p>\n

\u7531\u65bc \\(e\\) \u7e3d\u559c\u6b61\u85cf\u8eab\u81ea\u7136\u8207\u751f\u6d3b\u4e2d\uff0c\u6240\u4ee5\u6211\u5011\u5148\u4f86\u8003\u616e\u4e00\u500b\u8207\u8907\u5229\u6709\u95dc\u7684\u554f\u984c\uff1a\u5047\u8a2d\u672c\u91d1\u70ba \\(1\\) \u55ae\u4f4d\uff0c\u4e26\u4ee5\u8907\u5229\u7684\u65b9\u5f0f\u8a08\u7b97\u3002<\/p>\n

\u82e5\u5229\u7387\u70ba \\(100\\%\\)\uff0c\u90a3\u9ebc \\(1\\) \u5e74\u5f8c\u672c\u5229\u548c\u70ba \\((1+1)^1=2\\)\u3002<\/p>\n

\u82e5\u6539\u6210\u534a\u5e74\u652f\u4ed8\u4e00\u6b21\u5229\u606f\uff0c\u5247\u5229\u7387\u6e1b\u534a\u70ba \\(\\frac{1}{2}\\cdot 100\\%\\)\u3002
\n\u90a3\u9ebc\uff0c\\(1\\) \u5e74\u5f8c\u672c\u5229\u548c\u70ba\uff1a\\((1+\\frac{1}{2})^2=\\frac{9}{4}=2.25\\)<\/p>\n

\u82e5\u6539\u6210\u56db\u500b\u6708\u652f\u4ed8\u4e00\u6b21\u5229\u606f\uff0c\u5247\u5229\u7387\u8b8a\u70ba\u00a0\\(\\frac{1}{3}\\cdot 100\\%\\)\u3002
\n\u90a3\u9ebc\uff0c\\(1\\) \u5e74\u5f8c\u672c\u5229\u548c\u70ba\uff1a\\((1+\\frac{1}{3})^3=\\frac{64}{27}\\approx 2.370…\\)<\/p>\n

\u82e5\u6539\u6210\u4e09\u500b\u6708\u652f\u4ed8\u4e00\u6b21\u5229\u606f\uff0c\u5247\u5229\u7387\u8b8a\u70ba\u00a0\\(\\frac{1}{4}\\cdot 100\\%\\)\u3002
\n\u90a3\u9ebc\uff0c\\(1\\) \u5e74\u5f8c\u672c\u5229\u548c\u70ba\uff1a\\((1+\\frac{1}{4})^4=\\frac{625}{256}\\approx 2.441…\\)<\/p>\n

\\(\\cdots\\)<\/p>\n

\u4ee5\u6b64\u985e\u63a8\uff0c\u7576\u5229\u7387\u8b8a\u6210\u539f\u672c\u7684 \\(\\frac{1}{n}\\)\uff0c\u652f\u4ed8\u6b21\u6578\u8b8a\u6210 \\(n\\)\u00a0<\/i>\u6b21\uff0c\u5247 \\(1\\) \u5e74\u5f8c\u672c\u5229\u548c\u70ba\uff1a\\((1+\\frac{1}{n})^n\\)<\/p>\n

\u5982\u6b64\uff0c\u6211\u5011\u53ef\u6216\u5f97\u4e00\u500b\u6578\u5217\u3008\\((1+\\frac{1}{n})^n\\)\u3009\uff0c\u5176\u4e2d \\(n\\)\u00a0<\/i>\u70ba\u81ea\u7136\u6578\u3002\u00a0<\/p>\n

\u8a08\u7b97\u4e26\u89c0\u5bdf\u6578\u5217\u7684\u524d\u56db\u9805\u5f8c\u53ef\u767c\u73fe\uff1a\\(a_1=2,~a_2=2.25,~a_3=2.370…,~a_4=2.441…\\)<\/p>\n

\u4e5f\u8a31\u4f60\u6703\u731c\u6e2c\uff0c\u9019\u500b\u6578\u5217\u5404\u9805\u8d8a\u4f86\u8d8a\u5927\uff0c\u4f46\u5b83\u6709\u53ef\u80fd\u7121\u6b62\u76e1\u5730\u8b8a\u5927\u589e\u52a0\u55ce\uff1f<\/p>\n

\u82e5\u6211\u5011\u518d\u7e7c\u7e8c\u8a08\u7b97\u4e0b\u53bb\uff0c\u53ef\u4ee5\u767c\u73fe\u4e0b\u5217\u8fd1\u4f3c\u503c\uff1a<\/p>\n\n\n\n\n
\\(n\\)<\/td>\n10<\/td>\n100<\/td>\n1,000<\/td>\n10,000<\/td>\n100,000<\/td>\n1,000,000<\/td>\n<\/tr>\n
\\((1+\\frac{1}{n})^n\\)<\/td>\n2.59374<\/td>\n2.70481<\/td>\n2.71692<\/td>\n2.71815<\/td>\n2.71827<\/td>\n2.71828<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u773c\u5c16\u7684\u8b80\u8005\u53ef\u80fd\u6703\u767c\u73fe\uff0c\u5b83\u6210\u9577\u300c\u8b8a\u5927\u300d\u7684\u8da8\u52e2\u8d8a\u4f86\u8d8a\u7de9\u6162\uff0c\u7576 \\(n\\)\u00a0<\/i>\u5f88\u5927\u6642\uff0c\u5b83\u5e7e\u4e4e\u5feb\u8981\u300c\u4e0d\u52d5\u300d\u4e86\u3002\u63a5\u8457\uff0c\u8b80\u8005\u53ef\u80fd\u500b\u7136\u806f\u60f3\u5230\u4e0b\u4e00\u500b\u554f\u984c\uff1a\u5b83\u662f\u5426\u6709\u4e0a\u754c\uff1f\u518d\u8005\uff0c\u66f4\u9032\u4e00\u6b65\u5730\uff0c\u5b83\u662f\u5426\u6536\u6582\u5462\uff1f\u82e5\u6536\u6582\uff0c\u5b83\u7684\u6975\u9650\u503c\u53c8\u70ba\u4f55\u5462\uff1f\u4ee5\u4e0b\uff0c\u6211\u5011\u5206\u6790\u4e00\u4e9b\u8207\u6b64\u6578\u5217\u6709\u95dc\u7684\u6027\u8cea\uff0c\u5f15\u51fa\u81ea\u7136\u5c0d\u6578\u7684\u5e95\u6578 \\(e\\)\uff0c\u6700\u5f8c\u8b49\u660e\u5b83\u70ba\u7121\u7406\u6578\u3002<\/p>\n

\u70ba\u4e86\u8b49\u660e\u6b64\u6578\u5217\u7684\u6975\u9650\u503c\u5b58\u5728\uff0c\u6211\u5011\u5f15\u5165\u4e00\u500b\u8207\u5be6\u6578\u5b8c\u5099\u6027\u6709\u95dc\u5e38\u7528\u7684\u5224\u5225\u5b9a\u7406\u3002<\/p>\n

\u5b9a\u7406\uff1a\u8a2d\u7121\u7aae\u6578\u5217 \\(<a_n>\\)\u00a0\u905e\u589e\u4e14\u6709\u4e0a\u754c\uff0c\u5247\u6b64\u6578\u5217\u6536\u6582\u3002<\/b><\/p>\n

\u56e0\u6b64\uff0c\u6211\u5011\u9700\u8b49\u660e\uff1a(1)\u6b64\u6578\u5217\u905e\u589e\uff0c(2)\u6b64\u6578\u5217\u4e14\u6709\u4e0a\u754c\u3002
\n\u4f9d\u64da\u4e0a\u8ff0\u5b9a\u7406\uff0c\u5373\u53ef\u7531\u90192\u500b\u7d50\u679c\u7372\u5f97\u6b64\u6578\u5217\u6536\u6582\u7684\u7d50\u8ad6\u3002<\/p>\n

\u8b49\u660e\uff1a<\/p>\n

(1) \u6b64\u6578\u5217\u905e\u589e<\/strong><\/p>\n

\u8981\u8b49\u660e\u6578\u5217\u905e\u589e\uff0c\u4e0d\u96e3\u60f3\u50cf\u9700\u8981\u8b49\u660e\u5b83\u7684\u5f8c\u9805\u5927\u65bc\u7b49\u65bc\u524d\u9805\uff0c\u4ea6\u5373\u8b49\u660e\u5c0d\u6240\u6709\u7684\u81ea\u7136\u6578 \\(n\\)\uff0c\u7686\u6eff\u8db3 \\(a_n\\le a_{n+1}\\)\u3002\u5e38\u7528\u7684\u65b9\u6cd5\u5c31\u662f\u300c\u76f4\u63a5\u6bd4\u5927\u5c0f\u300d\uff0c\u600e\u9ebc\u6bd4\u5462\uff1f\u53ef\u4ee5\u62ff\u5f8c\u9805\u6e1b\u524d\u9805 \\(a_{n+1}-a_n\\)\uff0c\u4e26\u8b49\u660e\u5176\u503c\u6046\u70ba\u6b63\u6216 \\(0\\)\u3002\u53c8\u6216\u8005\uff0c\u62ff\u5f8c\u9805\u9664\u524d\u9805 \\(\\frac{a_{n+1}}{a_n}\\)\uff0c\u4e26\u8b49\u660e\u5176\u503c\u6046\u5927\u65bc\u7b49\u65bc \\(1\\)\u3002<\/p>\n

\u53ef\u60dc\u7684\u662f\uff0c\u91dd\u5c0d \\(<(1+\\frac{1}{n})^n>\\)\u00a0\u9019\u500b\u6578\u5217\u800c\u8a00\uff0c\u4ee5\u9019\u5169\u7a2e\u65b9\u6cd5\u4e26\u4e0d\u6613\u884c\uff08\u6709\u8208\u8da3\u7684\u8b80\u8005\u53ef\u81ea\u884c\u5617\u8a66\u770b\u770b\uff01\uff09\u3002\u56e0\u6b64\uff0c\u9019\u88e1\u9700\u8981\u65b0\u7684\u6bd4\u6cd5\uff1a\u300c\u9593\u63a5\u6bd4\u5927\u5c0f\u300d\u3002<\/p>\n

\u6211\u5011\u8f49\u500b\u5f4e\uff0c\u5229\u7528\u4e8c\u9805\u5f0f\u5b9a\u7406\uff0c\u5206\u5225\u5c07 \\(a_n\\)\u00a0\u8207 \\(a_{n+1}\\)\u00a0\u7d66\u300c\u6253\u958b\u300d\uff0c\u518d\u4f5c\u6bd4\u8f03\u3002
\n\u5c55\u958b\u8207\u6bd4\u8f03\u65b9\u5f0f\u5982\u4e0b\uff1a<\/p>\n

\\(\\begin{array}{ll}{a_n} &= {(1 + \\frac{1}{n})^n} = C_0^n + C_1^n(\\frac{1}{n}) + C_2^n{(\\frac{1}{n})^2}+\\cdots+ C_n^n{(\\frac{1}{n})^n}\\\\&= 1 + n(\\frac{1}{n})+\\frac{{n(n – 1)}}{{2!}}{(\\frac{1}{n})^2}+\\frac{{n(n – 1)(n – 2)}}{{3!}}{(\\frac{1}{n})^3} +\\cdots+ \\frac{{n!}}{{n!}}{(\\frac{1}{n})^n}\\\\&= 1 + 1 + \\frac{1}{{2!}}(1 – \\frac{1}{n}) + \\frac{1}{{3!}}(1 – \\frac{1}{n})(1 – \\frac{2}{n})+\\cdots+ \\frac{1}{{n!}}(1 – \\frac{1}{n})(1 – \\frac{2}{n}) \\cdots (1 – \\frac{{n – 1}}{n})\\end{array}\\)<\/p>\n

\\(\\begin{array}{ll}{a_{n + 1}} &= {(1 + \\frac{1}{{n + 1}})^{n + 1}}\\\\&= 1 + 1 + \\frac{1}{{2!}}(1 – \\frac{1}{{n + 1}}) + \\frac{1}{{3!}}(1 – \\frac{1}{{n + 1}})(1 – \\frac{2}{{n + 1}}) +\\cdots+ \\frac{1}{{n!}}(1 – \\frac{1}{{n + 1}})(1 – \\frac{2}{{n + 1}}) \\cdots (1 – \\frac{{n – 1}}{{n + 1}}) + \\frac{1}{{(n + 1)!}}(1 – \\frac{1}{{n + 1}})(1 – \\frac{2}{{n + 1}}) \\cdots (1 – \\frac{n}{{n + 1}}) \\end{array}\\)<\/p>\n

\u63a5\u8457\uff0c\u6bd4\u8f03 \\(a_n\\)\u00a0\u8207 \\(a_{n+1}\\)\u00a0\u524d \\(n\\)\u00a0\u9805\uff0c\u53ef\u77e5\uff1a<\/p>\n

\\(\\frac{1}{{k!}}(1 – \\frac{1}{k})(1 – \\frac{2}{k}) \\cdots (1 – \\frac{{k – 1}}{k}) \\le \\frac{1}{{k!}}(1 – \\frac{1}{{k + 1}})(1 – \\frac{2}{{k + 1}}) \\cdots (1 – \\frac{{k – 1}}{{k + 1}})\\)<\/p>\n

\uff0c\\(\\forall k=1,2,…,n\\)<\/p>\n

\u5c1a\u4e0d\u7136\u52a0\u4e0a \\)a_{n+1}\\(\u00a0\u7684\u6700\u5f8c\u4e00\u9805\uff0c\u6211\u5011\u5373\u8b49\u660e\u4e86\u5c0d\u6240\u6709\u7684\u81ea\u7136\u6578\uff0c\\)a_n\\le a_{n+1}\\(\u3002<\/p>\n

(2) \u6b64\u6578\u5217\u6709\u4e0a\u754c<\/strong><\/p>\n

\u6b32\u8b49\u660e\u6b64\u6578\u5217\u6709\u4e0a\u754c\u5373\u662f\u8b49\u660e\u5b58\u5728\u67d0\u500b\u300c\u5920\u5927\u7684\u6578\u300d\uff0c\u4f7f\u5f97\u6578\u5217\u7684\u6bcf\u4e00\u9805\u90fd\u6bd4\u9019\u500b\u6578\uff08\u4e0a\u754c\uff09\u5b83\u4f86\u5f97\u5c0f\u3002\u9019\u88e1\u6211\u5011\u540c\u6a23\u5229\u7528\u4e0a\u8ff0\u4e8c\u9805\u5f0f\u5b9a\u7406\u5c55\u958b\u5f8c\u7684\u7d50\u679c\u3002\u9996\u5148\uff0c\u56e0\u70ba\uff1a<\/p>\n

\\(\\begin{array}{ll}{(1 + \\frac{1}{n})^n} &= 1 + 1 + \\frac{1}{{2!}}(1 – \\frac{1}{n}) + \\frac{1}{{3!}}(1 – \\frac{1}{n})(1 – \\frac{2}{n}) +\\cdots+ \\frac{1}{{n!}}(1 – \\frac{1}{n})(1 – \\frac{2}{n}) \\cdots (1 – \\frac{{n – 1}}{n})\\end{array}\\)<\/p>\n

\u4e0d\u96e3\u770b\u51fa\uff1a\\(2 < {(1 + \\frac{1}{n})^n} < 2 + \\frac{1}{2} + \\frac{1}{4} +\\cdots+ \\frac{1}{{{2^{n – 1}}}} < 3\\)<\/p>\n

\u56e0\u6b64\uff0c\u6578\u5217 \\(<(1+\\frac{1}{n})^n>\\) \u6bcf\u4e00\u9805\u7684\u503c\u7684\u78ba\u88ab\u9650\u5236\u65bc \\(3\\) \u9019\u500b\u6578\u4e4b\u4e0b\u3002<\/p>\n

\u7531(1)\u8207(2)\u53ef\u77e5\u6578\u5217 \\(<(1+\\frac{1}{n})^n>\\)\u00a0\u905e\u589e\u4e14\u6709\u4e0a\u754c\uff0c\u6545\u5176\u6536\u6582\uff0c\u56e0\u6b64\u00a0\\(\\mathop {\\lim }\\limits_{n \\to \\infty } {(1 + \\frac{1}{n})^n}\\)\u00a0\u5b58\u5728\u3002\u65bc\u662f\uff0c\u6211\u5011\u4fbf\u628a\u6b64\u6975\u9650\u503c\u5b9a\u70ba \\(e\\)\uff0c\u5b83\u4ea6\u88ab\u7a31\u70ba\u81ea\u7136\u5c0d\u6578\u7684\u5e95\u6578\u3002\u63a5\u8457\uff0c\u5229\u7528\u6578\u5217 \\(<(1+\\frac{1}{n})^n>\\)\u00a0\u4fbf\u53ef\u8a08\u7b97\u51fa\u5176\u8fd1\u4f3c\u503c\uff1a<\/p>\n

\\(e = 2.7182818284 59…\\)<\/p>\n

\u81f3\u65bc\u5b83\u70ba\u4f55\u662f\u7121\u7406\u6578\uff0c\u7a0d\u5f8c\u4f5c\u8aaa\u660e\u3002<\/p>\n

\u4e0b\u5716\u4e00\u70ba\u51fd\u6578 \\(f(x)=(1+\\frac{1}{x})^x\\) \u7684\u5716\u5f62\u3002\u53d6 \\(x\\)\u00a0<\/i>\u70ba\u6574\u6578\u6642\uff0c\u4fbf\u662f\u4e0a\u8ff0\u7684\u6578\u5217 \\(<(1+\\frac{1}{n})^n>\\)\u3002
\n\u5f9e\u5716\u4e2d\u4ea6\u53ef\u770b\u51fa\u96a8\u8457 \\(x\\)\u00a0<\/i>\u8b8a\u5927\uff0c\u5b83\u7684\u51fd\u6578\u503c\u4e5f\u4e0d\u65b7\u8b8a\u5927\uff0c\u53ea\u662f\u4e00\u958b\u59cb\u4e0a\u5347\u7684\u901f\u5ea6\u8f03\u5feb\uff0c\u4f46\u6162\u6162\u5730\u4fbf\u958b\u59cb\u8da8\u7de9\u4e86\uff0c\u751a\u81f3\u7576 \\(x\\)\u00a0<\/i>\u5927\u65bc \\(10\\) \u4e4b\u5f8c\uff0c\u5f9e\u5716\u5f62\u770b\u8d77\u4f86\u5df2\u63a5\u8fd1\u6c34\u5e73\u7dda\u4e86\u3002\u5f9e\u5716\u4e2d\u4e5f\u53ef\u4ee5\u767c\u73fe\uff0c\u51fd\u6578\u7684\u503c\u7686\u5728 \\(2\\) \u8207 \\(3\\) \u4e4b\u9593\u3002<\/p>\n

\"56597_p1\"<\/a>

\u5716\u4e00\\(~~\\)\u51fd\u6578 \\(f(x)=(1+\\frac{1}{x})^x\\) \u7684\u5716\u5f62<\/p><\/div>\n

\u672c\u6587\u4e00\u958b\u59cb\uff0c\u6211\u5011\u5f9e\u8907\u5229\u7684\u4f8b\u5b50\u5f97\u5230\u4e86\u6578\u5217\u00a0\\(<(1+\\frac{1}{n})^n>\\)\uff0c
\n\u4e26\u8b49\u660e\u4e86\u5b83\u905e\u589e\u4e14\u6709\u4e0a\u754c\uff0c\\(\\mathop {\\lim }\\limits_{n \\to \\infty } {(1 + \\frac{1}{n})^n}\\) \u5b58\u5728\uff0c\u4e26\u628a\u6b64\u6975\u9650\u503c\u5b9a\u70ba \\(e\\)\u3002
\n<\/i>\u53e6\u4e00\u65b9\u9762\uff0c\u8b80\u8005\u53ef\u80fd\u6703\u5728\u67d0\u4e9b\u66f8\u4e2d\u767c\u73fe\uff0c\u6211\u5011\u4e5f\u6703\u628a\u300c\\(e\\)\u300d\u5b9a\u7fa9\u6210\uff1a<\/p>\n

\\({\\rm{e}}=\\sum\\limits_{k = 0}^\\infty{\\frac{1}{{k!}}}= \\frac{1}{{0!}} + \\frac{1}{{1!}} + \\frac{1}{{2!}} + \\frac{1}{{3!}} + \\frac{1}{{4!}} +\\cdots+ \\frac{1}{{n!}} +\\cdots\\)<\/p>\n

\u770b\u5230\u9019\u88e1\uff0c\u8b80\u8005\u53ef\u80fd\u6703\u597d\u5947\uff0c\u7a76\u7adf\u9019\u662f\u600e\u9ebc\u4e00\u56de\u4e8b\uff1f\u4e0d\u662f\u8aaa \\(e\\) \u662f\u7121\u7406\u6578\u55ce\uff1f\u600e\u9ebc\u5b83\u53ef\u4ee5\u5beb\u6210\u4e00\u9577\u4e32\u6709\u7406\u6578\u4e4b\u548c\uff1f\u9019\u88e1\uff0c\u6211\u5011\u9032\u4e00\u6b65\u5206\u6790\u7121\u7aae\u7d1a\u6578\u00a0\\(\\sum\\limits_{k = 0}^\\infty{\\frac{1}{{k!}}}\\)\u00a0\u76f8\u95dc\u6027\u8cea\u3002\u9996\u5148\uff0c\u6211\u5011\u540c\u6a23\u8b49\u660e\u5b83\u6536\u6582\uff0c\u63a5\u8457\uff0c\u8b49\u660e\u5b83\u6070\u7b49\u65bc\u524d\u9762\u63d0\u5230\u7684\u00a0\\(\\mathop {\\lim }\\limits_{n \\to \\infty } {(1 + \\frac{1}{n})^n}\\)\uff0c\u4e26\u8b49\u660e\u5b83\u70ba\u7121\u7406\u6578\u3002<\/p>\n

\u6536\u6582\u6027\u8b49\u660e\u5982\u4e0b\uff1a<\/p>\n

\u8003\u616e\u7531\u524d \\(n\\)\u00a0<\/i>\u9805\u548c\u5f62\u6210\u7684\u6578\u5217 \\(<\\sum\\limits_{k = 0}^n {\\frac{1}{{k!}}} >\\)\uff0c\\(n=1,2,3,…\\)<\/p>\n

(1)\u56e0\u70ba\u539f\u6578\u5217\u5404\u9805\u7686\u6b63\uff0c\u56e0\u6b64\uff0c\u6578\u5217 \\(<\\sum\\limits_{k = 0}^\\infty{\\frac{1}{{k!}}}>\\) \u986f\u7136\u905e\u589e\u3002<\/p>\n

(2)\u518d\u8005\uff0c\\(\\begin{array}{ll}\\sum\\limits_{k = 0}^\\infty{\\frac{1}{{k!}}}&= \\frac{1}{{0!}} + \\frac{1}{{1!}} + \\frac{1}{{2!}} + \\frac{1}{{3!}} + \\frac{1}{{4!}} + \\frac{1}{{5!}} +\\cdots\\\\&\\le 1 + \\frac{1}{1} + \\frac{1}{2} + \\frac{1}{4} + \\frac{1}{8} + \\frac{1}{{16}} + \\frac{1}{{32}} +\\cdots= 1 + \\frac{1}{{1 – {\\textstyle{1 \\over 2}}}} = 3\\end{array}\\)<\/p>\n

\u7531(1)(2)\u4ee5\u53ca\u524d\u6587\u63d0\u5230\u7684\u6536\u6582\u5224\u5225\u5b9a\u7406
\n\u53ef\u77e5 \\(\\sum\\limits_{k = 0}^\\infty{\\frac{1}{{k!}}}= \\frac{1}{{0!}} + \\frac{1}{{1!}} + \\frac{1}{{2!}} + \\frac{1}{{3!}} +\\cdots\\frac{1}{{n!}} +\\cdots\\)\u7684\u6975\u9650\u503c\u5b58\u5728\u3002<\/p>\n

\u81f3\u65bc\u600e\u9ebc\u8b49\u660e\u00a0\\(\\sum\\limits_{k = 0}^\\infty{\\frac{1}{{k!}}}\\)\u00a0\u8207 \\(\\mathop {\\lim }\\limits_{n \\to \\infty } {(1 + \\frac{1}{n})^n}\\)\u00a0\u76f8\u7b49\u5462\uff1f<\/p>\n

\u53ea\u8981\u80fd\u5206\u5225\u8b49\u660e \\(\\sum\\limits_{k = 0}^\\infty{\\frac{1}{{k!}}}\\le \\mathop {\\lim }\\limits_{n \\to \\infty } {(1 + \\frac{1}{n})^n}\\)\u00a0\u4ee5\u53ca\u00a0\\(\\sum\\limits_{k = 0}^\\infty{\\frac{1}{{k!}}}\\ge \\mathop {\\lim }\\limits_{n \\to \\infty } {(1 + \\frac{1}{n})^n}\\)\uff0c<\/p>\n

\u5373\u80fd\u8b49\u660e \\(\\sum\\limits_{k = 0}^\\infty{\\frac{1}{{k!}}}\\)\u00a0\u8207\u00a0\\(\\mathop {\\lim }\\limits_{n \\to \\infty } {(1 + \\frac{1}{n})^n}\\) \u76f8\u7b49\u3002<\/p>\n

\u9996\u5148\uff0c\u6211\u5011\u5148\u8b49\u00a0\\(\\sum\\limits_{k = 0}^\\infty{\\frac{1}{{k!}}}\\ge \\mathop {\\lim }\\limits_{n \\to \\infty } {(1 + \\frac{1}{n})^n}\\)\u3002<\/p>\n

\u6211\u5011\u62ff\\(\\sum\\limits_{k = 0}^\\infty{\\frac{1}{{k!}}}\\)\u00a0\u524d \\(n\\)\u00a0<\/i>\u9805\u548c\u6240\u5f62\u6210\u7684\u6578\u5217\u00a0\\(<\\sum\\limits_{k = 0}^n {\\frac{1}{{k!}}}>\\)\u00a0\u8207\u7121\u7aae\u6578\u5217 \\(<{(1 + \\frac{1}{n})^n}>\\)\u00a0\u4f5c\u6bd4\u8f03\u3002<\/p>\n

\u540c\u6a23\u5229\u7528\u4e8c\u9805\u5c55\u5f0f\u5b9a\u7406\uff0c\u5c55\u958b\u6574\u7406\u5f8c\u53ef\u5f97\uff1a<\/p>\n

\\(\\begin{array}{ll}{(1 + \\frac{1}{n})^n} &= C_0^n + C_1^n(\\frac{1}{n}) + C_2^n{(\\frac{1}{n})^2} +\\cdots+ C_n^n{(\\frac{1}{n})^n}\\\\&=1 + n(\\frac{1}{n}) + \\frac{{n(n – 1)}}{{2!}}{(\\frac{1}{n})^2} + \\frac{{n(n – 1)(n – 2)}}{{3!}}{(\\frac{1}{n})^3} +\\cdots+ \\frac{{n!}}{{n!}}{(\\frac{1}{n})^n}\\\\&\\le 1 + 1 + \\frac{{{n^2}}}{{2!}}{(\\frac{1}{n})^2} + \\frac{{{n^3}}}{{3!}}{(\\frac{1}{n})^3} + \\frac{{{n^4}}}{{3!}}{(\\frac{1}{n})^4} +\\cdots+ \\frac{{{n^n}}}{{n!}}{(\\frac{1}{n})^n}\\\\&=1 + \\frac{1}{{1!}} + \\frac{1}{{2!}} + \\frac{1}{{3!}} +\\cdots+ \\frac{1}{{n!}} +\\cdots\\end{array}\\)<\/p>\n

\u5373\u00a0\\({(1 + \\frac{1}{n})^n}\\le\\sum\\limits_{k = 0}^n{\\frac{1}{{k!}}}\\)<\/p>\n

\u53e6\u4e00\u65b9\u9762\uff0c\u6211\u5011\u4e5f\u53ef\u8b49\u660e\u00a0\\(\\sum\\limits_{k = 0}^\\infty{\\frac{1}{{k!}}}\\le\\mathop {\\lim }\\limits_{n\\to\\infty }{(1 + \\frac{1}{n})^n}\\)\uff0c\u9019\u90e8\u4efd\u7684\u8b49\u660e\u7559\u4f5c\u7fd2\u984c\uff0c\u7559\u5f85\u8b80\u8005\u81ea\u884c\u5617\u8a66\u3002<\/p>\n

\u7d9c\u5408\u4e0a\u8ff0\u7d50\u679c\uff0c\u7531\u65bc \\({(1 + \\frac{1}{n})^n}\\le\\sum\\limits_{k=0}^n {\\frac{1}{{k!}}}\\)\u00a0\u4e14 \\(\\mathop {\\lim }\\limits_{n\\to\\infty}{(1 + \\frac{1}{n})^n}\\ge\\sum\\limits_{k=0}^n {\\frac{1}{{k!}}}\\)\uff0c\u53ef\u63a8\u5f97\uff1a<\/p>\n

\\(\\mathop {\\lim }\\limits_{n \\to \\infty }{(1 +\\frac{1}{n})^n}=\\sum\\limits_{k = 0}^\\infty{\\frac{1}{{k!}}}= 1 + \\frac{1}{{1!}} + \\frac{1}{{2!}} + \\frac{1}{{3!}} +\\cdots+ \\frac{1}{{n!}}+\\cdots\\)<\/p>\n

\u56e0\u6b64\uff0c\u53ef\u5f97\uff1a<\/p>\n

\\({\\rm{e}} =\\mathop {\\lim }\\limits_{n \\to \\infty }{(1 + \\frac{1}{n})^n} = 1 + \\frac{1}{{1!}} + \\frac{1}{{2!}} + \\frac{1}{{3!}} +\\cdots+ \\frac{1}{{n!}} +\\cdots\\)<\/p>\n

\u8b49\u660e\u4e86\u4e0a\u8ff0\u7d50\u679c\u5f8c\uff0c\u6211\u5011\u4fbf\u53ef\u7528\u4e4b\u8b49\u660e \\(e\\) \u70ba\u7121\u7406\u6578\u3002\u56de\u60f3\u9ad8\u4e2d\u6559\u6750\uff08\u7b2c\u4e00\u518a\u7684\u9644\u9304\uff09\u4e2d\uff0c\u8b49\u660e \\(\\sqrt{2}\\) \u70ba\u7121\u7406\u6578\u6642\uff0c\u5229\u7528\u4e86\u6b78\u8b2c\u8b49\u6cd5\u3002\u4f9d\u6b64\u9748\u611f\u8207\u7d93\u9a57\uff0c\u9019\u88e1\u6211\u5011\u540c\u6a23\u5229\u7528\u6b78\u8b2c\u8b49\u6cd5\u4f86\u8b49\u660e \\(e\\) \u70ba\u7121\u7406\u6578\u3002<\/p>\n

\u8b49\u660e\uff1a<\/p>\n

\u9996\u5148\uff0c\u5047\u8a2d\u00a0\\({\\rm{e}} = 1 + \\frac{1}{{1!}} + \\frac{1}{{2!}} + \\frac{1}{{3!}} +\\cdots+ \\frac{1}{{n!}} +\\cdots\\)\u00a0\u70ba\u6709\u7406\u6578\uff0c<\/p>\n

\u4ee4\u5176\u70ba \\(\\frac{p}{q}\\)\uff0c\u5176\u4e2d \\(p\\)\u00a0<\/i>\u8207 \\(q\\)\u00a0<\/i>\u7686\u70ba\u4e92\u8cea\u7684\u6b63\u6574\u6578\u3002<\/p>\n

\u7531\u00a0\\(\\frac{p}{q} = \\sum\\limits_{k = 0}^\\infty{\\frac{1}{{k!}}}= 1 + \\frac{1}{{1!}} + \\frac{1}{{2!}} + \\frac{1}{{3!}} +\\cdots+ \\frac{1}{{n!}} +\\cdots\\)<\/p>\n

\u53ef\u77e5\uff1a\\(0 < \\frac{p}{q} – (1 + \\frac{1}{{1!}} + \\frac{1}{{2!}} + \\frac{1}{{3!}} +\\cdots+ \\frac{1}{{q!}}) \\le \\frac{1}{{(q + 1)!}}[1 + \\frac{1}{{q + 1}} + {(\\frac{1}{{q + 1}})^2} +\\cdots] = \\frac{1}{{(q + 1)!}}\\frac{{q + 1}}{q} = \\frac{1}{{q \\cdot q!}}\\)<\/p>\n

\u540c\u4e58\u4ee5 \\(q{!}\\)\u00a0\u53ef\u5f97\uff1a\\(0 < q![\\frac{p}{q} – (1 + \\frac{1}{{1!}} + \\frac{1}{{2!}} + \\frac{1}{{3!}} +\\cdots+ \\frac{1}{{q!}})] \\le q!\\frac{1}{{q \\cdot q!}} = \\frac{1}{q} < 1\\)<\/p>\n

\u7136\u800c\uff0c\\(q![\\frac{p}{q} – (1 + \\frac{1}{{1!}} + \\frac{1}{{2!}} + \\frac{1}{{3!}} +\\cdots+ \\frac{1}{{q!}})] \\\\= q!(\\frac{p}{q}) – q!(1 + \\frac{1}{{1!}} + \\frac{1}{{2!}} + \\frac{1}{{3!}} +\\cdots+ \\frac{1}{{q!}})\\)<\/p>\n

\u5176\u4e2d\uff0c\\(q{!}(\\frac{p}{q})\\)\u00a0\u8207\u00a0\\(q!(1 + \\frac{1}{{1!}} + \\frac{1}{{2!}} + \\frac{1}{{3!}} +\\cdots+ \\frac{1}{{q!}})\\)\u00a0\u7686\u70ba\u81ea\u7136\u6578\u3002<\/p>\n

\u4f46\u662f\uff0c\u4e26\u4e0d\u5b58\u5728\u4ecb\u65bc \\(0\\) \u8207 \\(1\\) \u4e4b\u9593\u7684\u81ea\u7136\u6578\uff0c\u56e0\u6b64\u5c0e\u81f4\u4e86\u77db\u76fe\u3002\u800c\u9020\u6210\u6b64\u77db\u76fe\u7684\u539f\u56e0\uff0c\u662f\u56e0\u70ba\u6211\u5011\u5047\u8a2d \\(e\\) \u70ba\u6709\u7406\u6578\uff0c\u65bc\u662f \\(e\\) \u7576\u70ba\u7121\u7406\u6578\u624d\u662f\uff0c\u8b49\u660e\u5b8c\u6210\uff01<\/p>\n

\u7d9c\u5408\u672c\u6587\u5167\u5bb9\uff0c\u6211\u5011\u63d0\u51fa\u5e7e\u500b\u8207 \\(e\\) \u6709\u95dc\u7684\u7d50\u8ad6\uff1a<\/p>\n

    \n
  1. \\({\\rm{e}} = \\sum\\limits_{k = 0}^\\infty{\\frac{1}{{k!}}}= \\frac{1}{{0!}} + \\frac{1}{{1!}} + \\frac{1}{{2!}} + \\frac{1}{{3!}} +\\cdots+ \\frac{1}{{n!}} +\\cdots\\)\u3002<\/li>\n
  2. \\({\\rm{e}} = \\mathop {\\lim }\\limits_{n \\to \\infty }{(1 + \\frac{1}{n})^n}\\)\u3002<\/li>\n
  3. \\(e\\) \u70ba\u7121\u7406\u6578\u3002<\/li>\n
  4. \u00a0\\(2<e<3\\)\u3002<\/li>\n<\/ol>\n

    \u9664\u4e86\u4e0a\u8ff0\u7bc4\u570d\u5916\uff0c\u5be6\u969b\u4e0a\u6211\u5011\u4e5f\u53ef\u5229\u7528\u6578\u5217 \\(<{(1 + \\frac{1}{n})^n}>\\)
    \n\u6216\u5229\u7528\u00a0\\(1 + \\frac{1}{{1!}} + \\frac{1}{{2!}} + \\frac{1}{{3!}} +\\cdots+ \\frac{1}{{n!}} +\\cdots\\)\u00a0\u4f86\u8a08\u7b97\u5176\u8fd1\u4f3c\u503c\uff1a<\/p>\n

    \\(e = 2.718281828459…\\)<\/p>\n

    \u6b64\u5373\u70ba\u5713\u5468\u7387\u4e4b\u5916\uff0c\u53e6\u4e00\u500b\u91cd\u8981\u7684\u7121\u7406\u6578\u3002\u5176\u5c0f\u6578\u9ede\u5f8c\u7684\u6578\u5b57\u7121\u7aae\u7121\u76e1\uff0c\u4e0d\u5faa\u74b0\u4ea6\u7121\u898f\u5f8b\uff0c\u85cf\u8eab\u5927\u81ea\u7136\u3002\u4ee5\u5176\u6240\u9020\u51fa\u7684\u6307\u6578\u51fd\u6578 \\(e^x\\)\u00a0\u4ea6\u70ba\u5e38\u898b\u4e14\u91cd\u8981\u7684\u51fd\u6578\u3002<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

    \u53e6\u4e00\u500b\u91cd\u8981\u7684\u7121\u7406\u6578\uff1ae \uff08Another important irrational number\uff1ae\uff09 \u81fa\u5317\u5e02…<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[111,216],"tags":[7042,288,7043],"class_list":["post-56597","post","type-post","status-publish","format-standard","hentry","category-mathematics00","category-math02","tag-e","tag-288","tag-7043","loop-entry","cat-111","cat-216","no-thumbnail"],"views":14887,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/56597","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=56597"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/56597\/revisions"}],"predecessor-version":[{"id":86885,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/56597\/revisions\/86885"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=56597"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=56597"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=56597"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}