{"id":112631,"date":"2026-10-04T09:49:21","date_gmt":"2026-10-04T01:49:21","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/112631.html"},"modified":"2026-10-04T09:49:21","modified_gmt":"2026-10-04T01:49:21","slug":"18%e3%80%81%e3%80%90%e6%95%b0%e5%ad%a6%e3%80%91%e3%80%90%e5%9f%ba%e7%a1%80%e3%80%91%e6%b3%b0%e5%8b%92%e7%ba%a7%e6%95%b0%e6%80%8e%e4%b9%88%e6%8e%a8%e5%87%ba%e6%9d%a5%e7%9a%84%ef%bc%9a%e9%80%90%e9%98%b6","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/112631.html","title":{"rendered":"18\u3001\u3010\u6570\u5b66\u3011\u3010\u57fa\u7840\u3011\u6cf0\u52d2\u7ea7\u6570\u600e\u4e48\u63a8\u51fa\u6765\u7684\uff1a\u9010\u9636\u5339\u914d\u4e0e\u4f59\u9879"},"content":{"rendered":"

\u3010\u58f0\u660e\u3011\u672c\u535a\u5ba2\u6240\u6709\u5185\u5bb9\u5747\u4e3a\u4e2a\u4eba\u4e1a\u4f59\u65f6\u95f4\u521b\u4f5c，\u6240\u8ff0\u6280\u672f\u6848\u4f8b\u5747\u6765\u81ea\u516c\u5f00\u5f00\u6e90\u9879\u76ee（\u5982Github，Apache\u57fa\u91d1\u4f1a），\u4e0d\u6d89\u53ca\u4efb\u4f55\u4f01\u4e1a\u673a\u5bc6\u6216\u672a\u516c\u5f00\u6280\u672f，\u5982\u6709\u4fb5\u6743\u8bf7\u8054\u7cfb\u5220\u9664<\/p>\n

\u6807\u9898<\/h2>\n

18\u3001\u3010\u6570\u5b66\u3011\u3010\u57fa\u7840\u3011\u6cf0\u52d2\u7ea7\u6570\u600e\u4e48\u63a8\u51fa\u6765\u7684：\u9010\u9636\u5339\u914d\u4e0e\u4f59\u9879<\/p>\n

\u80cc\u666f<\/h2>\n

\u4e0a\u7bc7 blog \u3010\u6570\u5b66\u3011\u3010\u57fa\u7840\u3011\u6cf0\u52d2\u7ea7\u6570\u662f\u4ec0\u4e48：\u7528\u591a\u9879\u5f0f\u903c\u8fd1\u51fd\u6570 \u7ed9\u51fa\u4e86\u6cf0\u52d2\u7ea7\u6570\u4e0e\u9ea6\u514b\u52b3\u6797\u7ea7\u6570\u7684\u6837\u5b50： <\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

\u2211<\/p>\n

n<\/p>\n

=<\/p>\n

0<\/p>\n

\u221e<\/p>\n

f<\/p>\n

(<\/p>\n

n<\/p>\n

)<\/p>\n

(<\/p>\n

0<\/p>\n

)<\/p>\n

n<\/p>\n

!<\/p>\n

\u2009<\/p>\n

x<\/p>\n

n<\/p>\n

f(x)=\\\\sum_{n=0}^{\\\\infty}\\\\frac{f^{(n)}(0)}{n!}\\\\,x^{n}<\/p>\n

<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>n<\/span>=<\/span>0<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>\u221e<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>!<\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span><\/span>(<\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5e76\u8bb2\u6e05\u4e86\u4e09\u4e2a\u540d\u5b57（\u6cf0\u52d2\u591a\u9879\u5f0f\u662f\u90e8\u5206\u548c\u3001\u6cf0\u52d2\u7ea7\u6570\u662f\u6781\u9650\u3001\u9ea6\u514b\u52b3\u6797\u7ea7\u6570\u662f\u539f\u70b9\u5c55\u5f00）\u3001\u51e0\u4e2a\u5e38\u89c1\u5c55\u5f00，\u4ee5\u53ca"\u5c55\u5f00\u53ea\u5728\u6536\u655b\u534a\u5f84\u5185\u6709\u6548"\u3002\u4f46\u5b83\u7559\u4e86\u4e00\u4e2a\u6d1e：\u7cfb\u6570 <\/p>\n

f<\/p>\n

(<\/p>\n

n<\/p>\n

)<\/p>\n

(<\/p>\n

0<\/p>\n

)<\/p>\n

n<\/p>\n

!<\/p>\n

\\\\dfrac{f^{(n)}(0)}{n!}<\/p>\n

<\/span><\/span><\/span><\/span>n<\/span>!<\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span><\/span>(<\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u4ece\u54ea\u6765\u7684？ \u4e0a\u7bc7\u76f4\u63a5\u628a\u7ed3\u8bba\u7528\u4e86，\u672c\u7bc7\u628a\u5b83\u63a8\u51fa\u6765\u2014\u2014\u800c\u4e14\u987a\u624b\u8bf4\u6e05"\u4e3a\u4ec0\u4e48\u53ef\u4ee5\u771f\u7684\u7b49\u4e8e\u539f\u51fd\u6570"\u3002<\/p>\n

\n

\u76ee\u6807：\u8bbe\u51fa\u7cfb\u6570，\u518d\u628a\u5b83"\u89e3"\u51fa\u6765<\/h2>\n

\u5148\u5047\u8bbe\u51fd\u6570 <\/p>\n

f<\/p>\n

f<\/p>\n

<\/span><\/span>f<\/span><\/span><\/span><\/span><\/span> \u5728\u539f\u70b9\u9644\u8fd1\u53ef\u4ee5\u5199\u6210\u4e00\u4e32\u5e42\u4e4b\u548c（\u8fd9\u5728\u7b2c 17 \u7bc7\u7684\u6536\u655b\u8303\u56f4\u5185\u6210\u7acb）：<\/p>\n

<\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

a<\/p>\n

0<\/p>\n

+<\/p>\n

a<\/p>\n

1<\/p>\n

x<\/p>\n

+<\/p>\n

a<\/p>\n

2<\/p>\n

x<\/p>\n

2<\/p>\n

+<\/p>\n

a<\/p>\n

3<\/p>\n

x<\/p>\n

3<\/p>\n

+<\/p>\n

\u22ef<\/p>\n

f(x)=a_0+a_1x+a_2x^{2}+a_3x^{3}+\\\\cdots<\/p>\n

<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>+<\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>a<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u73b0\u5728\u7684\u4efb\u52a1\u662f：\u628a\u8fd9\u4e32\u7cfb\u6570 <\/p>\n

a<\/p>\n

0<\/p>\n

,<\/p>\n

a<\/p>\n

1<\/p>\n

,<\/p>\n

a<\/p>\n

2<\/p>\n

,<\/p>\n

\u2026<\/p>\n

a_0,a_1,a_2,\\\\dots<\/p>\n

<\/span><\/span>a<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>a<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>a<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span><\/span><\/span><\/span> \u7528\u51fd\u6570\u672c\u8eab\u7684\u4fe1\u606f\u786e\u5b9a\u4e0b\u6765\u3002\u601d\u8def\u53ea\u6709\u4e00\u62db\u2014\u2014\u9010\u9636\u6c42\u5bfc\u3002<\/p>\n

\n

\u9010\u9636\u6c42\u5bfc：\u6bcf\u6c42\u4e00\u6b21，\u5c31"\u66b4\u9732"\u4e00\u4e2a\u7cfb\u6570<\/h2>\n

\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/p>\n

\u56fe 1 \u662f\u8fd9\u4e2a\u6d88\u5143\u8fc7\u7a0b\u7684\u753b\u9762\u3002\u628a <\/p>\n

x<\/p>\n

=<\/p>\n

0<\/p>\n

x=0<\/p>\n

<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span> \u4ee3\u8fdb\u53bb，\u540e\u9762\u5e26 <\/p>\n

x<\/p>\n

x<\/p>\n

<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span> \u7684\u9879\u5168\u6ca1\u4e86：<\/p>\n

<\/p>\n

f<\/p>\n

(<\/p>\n

0<\/p>\n

)<\/p>\n

=<\/p>\n

a<\/p>\n

0<\/p>\n

f(0)=a_0<\/p>\n

<\/span><\/span>f<\/span>(<\/span>0<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u63a5\u7740\u5bf9\u4e24\u8fb9\u6c42\u5bfc\u3002<\/p>\n

a<\/p>\n

0<\/p>\n

a_0<\/p>\n

<\/span><\/span>a<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u5e38\u6570\u3001\u6c42\u5bfc\u53d8 <\/p>\n

0<\/p>\n

0<\/p>\n

<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>；<\/p>\n

a<\/p>\n

1<\/p>\n

x<\/p>\n

a_1x<\/p>\n

<\/span><\/span>a<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span><\/span><\/span><\/span> \u6c42\u5bfc\u5f97 <\/p>\n

a<\/p>\n

1<\/p>\n

a_1<\/p>\n

<\/span><\/span>a<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>；\u66f4\u9ad8\u6b21\u9879\u4ecd\u5e26 <\/p>\n

x<\/p>\n

x<\/p>\n

<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>：<\/p>\n

<\/p>\n

f<\/p>\n

\u2032<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

a<\/p>\n

1<\/p>\n

+<\/p>\n

2<\/p>\n

a<\/p>\n

2<\/p>\n

x<\/p>\n

+<\/p>\n

3<\/p>\n

a<\/p>\n

3<\/p>\n

x<\/p>\n

2<\/p>\n

+<\/p>\n

\u22ef<\/p>\n

\u27f9<\/p>\n

f<\/p>\n

\u2032<\/p>\n

(<\/p>\n

0<\/p>\n

)<\/p>\n

=<\/p>\n

a<\/p>\n

1<\/p>\n

f'(x)=a_1+2a_2x+3a_3x^{2}+\\\\cdots\\\\quad\\\\Longrightarrow\\\\quad f'(0)=a_1<\/p>\n

<\/span><\/span>f<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>2<\/span>a<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>+<\/span><\/span><\/span><\/span>3<\/span>a<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span>\u27f9<\/span><\/span><\/span><\/span><\/span>f<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u518d\u6c42\u4e00\u6b21\u5bfc，<\/p>\n

a<\/p>\n

1<\/p>\n

a_1<\/p>\n

<\/span><\/span>a<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e5f\u6ca1\u4e86，\u8f6e\u5230 <\/p>\n

2<\/p>\n

a<\/p>\n

2<\/p>\n

2a_2<\/p>\n

<\/span><\/span>2<\/span>a<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u9732\u5934：<\/p>\n

<\/p>\n

f<\/p>\n

\u2032<\/p>\n

\u2032<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

2<\/p>\n

a<\/p>\n

2<\/p>\n

+<\/p>\n

6<\/p>\n

a<\/p>\n

3<\/p>\n

x<\/p>\n

+<\/p>\n

\u22ef<\/p>\n

\u27f9<\/p>\n

f<\/p>\n

\u2032<\/p>\n

\u2032<\/p>\n

(<\/p>\n

0<\/p>\n

)<\/p>\n

=<\/p>\n

2<\/p>\n

a<\/p>\n

2<\/p>\n

f''(x)=2a_2+6a_3x+\\\\cdots\\\\quad\\\\Longrightarrow\\\\quad f''(0)=2a_2<\/p>\n

<\/span><\/span>f<\/span><\/span>\u2032\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span>a<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>6<\/span>a<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>+<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span>\u27f9<\/span><\/span><\/span><\/span><\/span>f<\/span><\/span>\u2032\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span>a<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5982\u6b64\u7ee7\u7eed，\u6bcf\u6c42\u4e00\u6b21\u5bfc\u3001\u6bcf\u4ee3\u4e00\u6b21 <\/p>\n

x<\/p>\n

=<\/p>\n

0<\/p>\n

x=0<\/p>\n

<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>，\u5c31"\u780d\u6389"\u524d\u9762\u6240\u6709\u9879\u3001\u53ea\u7559\u4e0b\u4e0b\u4e00\u4e2a\u7cfb\u6570（\u4e58\u4e0a\u4e00\u4e2a\u7cfb\u6570）\u3002\u89c4\u5f8b\u5f88\u6574\u9f50：<\/p>\n

<\/p>\n

f<\/p>\n

(<\/p>\n

n<\/p>\n

)<\/p>\n

(<\/p>\n

0<\/p>\n

)<\/p>\n

=<\/p>\n

n<\/p>\n

!
\n \u2009<\/p>\n

a<\/p>\n

n<\/p>\n

f^{(n)}(0)=n!\\\\,a_n<\/p>\n

<\/span><\/span>f<\/span><\/span>(<\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>n<\/span>!<\/span><\/span>a<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u628a\u5b83\u89e3\u51fa\u6765，\u5c31\u5f97\u5230\u6cf0\u52d2\u7cfb\u6570\u7684\u516c\u5f0f：<\/p>\n

<\/p>\n

a<\/p>\n

n<\/p>\n

=<\/p>\n

f<\/p>\n

(<\/p>\n

n<\/p>\n

)<\/p>\n

(<\/p>\n

0<\/p>\n

)<\/p>\n

n<\/p>\n

!<\/p>\n

a_n=\\\\frac{f^{(n)}(0)}{n!}<\/p>\n

<\/span><\/span>a<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>!<\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span><\/span>(<\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/p>\n

\u56fe 2 \u628a\u8fd9\u6761\u5bf9\u5e94\u5173\u7cfb\u5217\u5168：<\/p>\n

n<\/p>\n

n<\/p>\n

<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u9636\u5bfc\u6570\u5728 <\/p>\n

0<\/p>\n

0<\/p>\n

<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span> \u5904\u7684\u503c，\u9664\u4ee5 <\/p>\n

n<\/p>\n

!<\/p>\n

n!<\/p>\n

<\/span><\/span>n<\/span>!<\/span><\/span><\/span><\/span><\/span>，\u5c31\u662f\u7b2c <\/p>\n

n<\/p>\n

n<\/p>\n

<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u9879\u7cfb\u6570\u3002 \u7cfb\u6570\u88ab\u5404\u9636\u5bfc\u6570\u552f\u4e00\u786e\u5b9a\u2014\u2014\u8fd9\u5c31\u662f\u4e0a\u7bc7\u90a3\u4e2a\u516c\u5f0f\u7684\u6765\u5386\u3002<\/p>\n

\u987a\u5e26\u8bf4\u4e00\u53e5：\u6b63\u56e0\u4e3a\u7cfb\u6570\u88ab\u9489\u6b7b，\u8fd9\u6837\u7684\u5e42\u7ea7\u6570\u5c55\u5f00\u662f\u552f\u4e00\u7684\u2014\u2014\u4e00\u4e2a\u51fd\u6570\u5728\u4e00\u70b9\u5904\u53ea\u53ef\u80fd\u6709\u4e00\u7ec4\u8fd9\u6837\u7684\u7cfb\u6570\u3002<\/p>\n

\n

\u4e3a\u4ec0\u4e48\u504f\u504f\u9664\u4ee5 <\/p>\n

n<\/p>\n

!<\/p>\n

n!<\/p>\n

<\/span><\/span>n<\/span>!<\/span><\/span><\/span><\/span><\/span><\/h2>\n

\u5173\u952e\u5728\u6c42\u5bfc\u4f1a"\u964d\u5e42\u5e76\u5e26\u51fa\u7cfb\u6570"：<\/p>\n

x<\/p>\n

n<\/p>\n

x^{n}<\/p>\n

<\/span><\/span>x<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u8fde\u6c42 <\/p>\n

n<\/p>\n

n<\/p>\n

<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u6b21\u5bfc，\u4f1a\u628a\u6307\u6570\u4e00\u8def\u964d\u5230 <\/p>\n

0<\/p>\n

0<\/p>\n

<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>，\u540c\u65f6\u628a <\/p>\n

n<\/p>\n

\u22c5<\/p>\n

(<\/p>\n

n<\/p>\n

\u2212<\/p>\n

1<\/p>\n

)<\/p>\n

\u22ef<\/p>\n

1<\/p>\n

=<\/p>\n

n<\/p>\n

!<\/p>\n

n\\\\cdot(n-1)\\\\cdots1=n!<\/p>\n

<\/span><\/span>n<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>(<\/span>n<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span>)<\/span><\/span>\u22ef<\/span><\/span>1<\/span><\/span>=<\/span><\/span><\/span><\/span>n<\/span>!<\/span><\/span><\/span><\/span><\/span> \u8fd9\u4e2a\u56e0\u5b50\u5e26\u5230\u524d\u9762：<\/p>\n

<\/p>\n

(<\/p>\n

x<\/p>\n

n<\/p>\n

)<\/p>\n

(<\/p>\n

n<\/p>\n

)<\/p>\n

=<\/p>\n

n<\/p>\n

!<\/p>\n

\\\\bigl(x^{n}\\\\bigr)^{(n)}=n!<\/p>\n

<\/span><\/span>(<\/span><\/span>x<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span>(<\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>n<\/span>!<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5\u7b2c <\/p>\n

n<\/p>\n

n<\/p>\n

<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u9879 <\/p>\n

a<\/p>\n

n<\/p>\n

x<\/p>\n

n<\/p>\n

a_nx^{n}<\/p>\n

<\/span><\/span>a<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5728\u6c42\u5bfc <\/p>\n

n<\/p>\n

n<\/p>\n

<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u6b21\u540e\u53d8\u6210 <\/p>\n

n<\/p>\n

!
\n \u2009<\/p>\n

a<\/p>\n

n<\/p>\n

n!\\\\,a_n<\/p>\n

<\/span><\/span>n<\/span>!<\/span><\/span>a<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u800c\u5b83\u5fc5\u987b\u7b49\u4e8e <\/p>\n

f<\/p>\n

(<\/p>\n

n<\/p>\n

)<\/p>\n

(<\/p>\n

0<\/p>\n

)<\/p>\n

f^{(n)}(0)<\/p>\n

<\/span><\/span>f<\/span><\/span>(<\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002\u9664\u4ee5 <\/p>\n

n<\/p>\n

!<\/p>\n

n!<\/p>\n

<\/span><\/span>n<\/span>!<\/span><\/span><\/span><\/span><\/span>，\u6b63\u662f\u4e3a\u4e86\u62b5\u6d88\u8fd9\u4e2a\u964d\u5e42\u5e26\u51fa\u6765\u7684\u56e0\u5b50\u3002<\/p>\n

\n

\u6362\u6210\u4e00\u822c\u70b9：\u628a <\/p>\n

x<\/p>\n

x<\/p>\n

<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span> \u6362\u6210 <\/p>\n

(<\/p>\n

x<\/p>\n

\u2212<\/p>\n

a<\/p>\n

)<\/p>\n

(x-a)<\/p>\n

<\/span><\/span>(<\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>a<\/span>)<\/span><\/span><\/span><\/span><\/span><\/h2>\n

\u5728\u70b9 <\/p>\n

a<\/p>\n

a<\/p>\n

<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u5904\u5c55\u5f00\u65f6，\u628a\u4e0a\u9762\u7684 <\/p>\n

x<\/p>\n

x<\/p>\n

<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span> \u90fd\u66ff\u6362\u6210 <\/p>\n

(<\/p>\n

x<\/p>\n

\u2212<\/p>\n

a<\/p>\n

)<\/p>\n

(x-a)<\/p>\n

<\/span><\/span>(<\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>a<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u63a8\u5bfc\u5b8c\u5168\u4e00\u6837：<\/p>\n

<\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

a<\/p>\n

0<\/p>\n

+<\/p>\n

a<\/p>\n

1<\/p>\n

(<\/p>\n

x<\/p>\n

\u2212<\/p>\n

a<\/p>\n

)<\/p>\n

+<\/p>\n

a<\/p>\n

2<\/p>\n

(<\/p>\n

x<\/p>\n

\u2212<\/p>\n

a<\/p>\n

)<\/p>\n

2<\/p>\n

+<\/p>\n

\u22ef<\/p>\n

\u27f9<\/p>\n

a<\/p>\n

n<\/p>\n

=<\/p>\n

f<\/p>\n

(<\/p>\n

n<\/p>\n

)<\/p>\n

(<\/p>\n

a<\/p>\n

)<\/p>\n

n<\/p>\n

!<\/p>\n

f(x)=a_0+a_1(x-a)+a_2(x-a)^{2}+\\\\cdots\\\\quad\\\\Longrightarrow\\\\quad a_n=\\\\frac{f^{(n)}(a)}{n!}<\/p>\n

<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>a<\/span>)<\/span><\/span>+<\/span><\/span><\/span><\/span>a<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>a<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span>\u27f9<\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>!<\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span><\/span>(<\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>a<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e8e\u662f\u5f97\u5230\u4e00\u822c\u5f62\u5f0f\u7684\u6cf0\u52d2\u7ea7\u6570：<\/p>\n

<\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

\u2211<\/p>\n

n<\/p>\n

=<\/p>\n

0<\/p>\n

\u221e<\/p>\n

f<\/p>\n

(<\/p>\n

n<\/p>\n

)<\/p>\n

(<\/p>\n

a<\/p>\n

)<\/p>\n

n<\/p>\n

!<\/p>\n

\u2009<\/p>\n

(<\/p>\n

x<\/p>\n

\u2212<\/p>\n

a<\/p>\n

)<\/p>\n

n<\/p>\n

f(x)=\\\\sum_{n=0}^{\\\\infty}\\\\frac{f^{(n)}(a)}{n!}\\\\,(x-a)^{n}<\/p>\n

<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>n<\/span>=<\/span>0<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>\u221e<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>!<\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span><\/span>(<\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>a<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>a<\/span>)<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/p>\n

a<\/p>\n

=<\/p>\n

0<\/p>\n

a=0<\/p>\n

<\/span><\/span>a<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span> \u65f6\u9000\u5316\u4e3a\u9ea6\u514b\u52b3\u6797\u7ea7\u6570\u3002\u201c\u5c55\u5f00\u70b9"\u5c31\u662f"\u5bfc\u6570\u7684\u53d6\u503c\u70b9\u201d\u2014\u2014\u8fd9\u4e5f\u89e3\u91ca\u4e86\u4e3a\u4ec0\u4e48\u79bb\u5c55\u5f00\u70b9\u8d8a\u8fd1\u3001\u903c\u8fd1\u8d8a\u597d\u3002<\/p>\n

\n

\u66f4\u4e25\u8c28\u7684\u4e00\u6b65：\u4f59\u9879，\u624d\u51b3\u5b9a"\u662f\u5426\u7b49\u4e8e\u539f\u51fd\u6570"<\/h2>\n

\u5230\u8fd9\u91cc，\u53ea\u8bc1\u660e\u4e86"\u5982\u679c\u51fd\u6570\u80fd\u5199\u6210\u5e42\u7ea7\u6570，\u90a3\u7cfb\u6570\u5fc5\u987b\u957f\u8fd9\u6837"\u3002\u4f46\u53cd\u8fc7\u6765\u2014\u2014\u7528\u8fd9\u4e9b\u7cfb\u6570\u62fc\u51fa\u6765\u7684\u7ea7\u6570，\u5c31\u4e00\u5b9a\u7b49\u4e8e\u539f\u51fd\u6570\u5417？ \u4e0d\u4e00\u5b9a\u3002\u771f\u6b63\u51b3\u5b9a\u8fd9\u4ef6\u4e8b\u7684，\u662f\u4f59\u9879（remainder）\u3002<\/p>\n

\u628a\u51fd\u6570\u5199\u6210"\u524d <\/p>\n

N<\/p>\n

N<\/p>\n

<\/span><\/span>N<\/span><\/span><\/span><\/span><\/span> \u9879\u591a\u9879\u5f0f + \u5269\u4e0b\u7684\u5c3e\u5df4"：<\/p>\n

<\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

S<\/p>\n

N<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

\u23df<\/p>\n

\u6cf0\u52d2\u591a\u9879\u5f0f<\/p>\n

+<\/p>\n

R<\/p>\n

N<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

\u23df<\/p>\n

\u4f59\u9879<\/p>\n

f(x)=\\\\underbrace{S_N(x)}_{\\\\text{\u6cf0\u52d2\u591a\u9879\u5f0f}}+\\\\underbrace{R_N(x)}_{\\\\text{\u4f59\u9879}}<\/p>\n

<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u6cf0\u52d2\u591a\u9879\u5f0f<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/p>\n

<\/span><\/p>\n

<\/span><\/span><\/span><\/span>S<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span>\u4f59\u9879<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/p>\n

<\/span><\/p>\n

<\/span><\/span><\/span><\/span>R<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4f59\u9879\u6709\u4e00\u4e2a\u5e38\u7528\u7684\u4f30\u8ba1（\u62c9\u683c\u6717\u65e5\u4f59\u9879，Lagrange remainder）：<\/p>\n

<\/p>\n

R<\/p>\n

N<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

f<\/p>\n

(<\/p>\n

N<\/p>\n

+<\/p>\n

1<\/p>\n

)<\/p>\n

(<\/p>\n

\u03be<\/p>\n

)<\/p>\n

(<\/p>\n

N<\/p>\n

+<\/p>\n

1<\/p>\n

)<\/p>\n

!<\/p>\n

\u2009<\/p>\n

(<\/p>\n

x<\/p>\n

\u2212<\/p>\n

a<\/p>\n

)<\/p>\n

N<\/p>\n

+<\/p>\n

1<\/p>\n

,<\/p>\n

\u03be<\/p>\n

\u00a0\u4ecb\u4e8e\u00a0<\/p>\n

a<\/p>\n

\u00a0\u4e0e\u00a0<\/p>\n

x<\/p>\n

\u00a0\u4e4b\u95f4<\/p>\n

R_N(x)=\\\\frac{f^{(N+1)}(\\\\xi)}{(N+1)!}\\\\,(x-a)^{N+1},\\\\qquad \\\\xi\\\\ \\\\text{\u4ecb\u4e8e}\\\\ a\\\\ \\\\text{\u4e0e}\\\\ x\\\\ \\\\text{\u4e4b\u95f4}<\/p>\n

<\/span><\/span>R<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>N<\/span><\/span>+<\/span><\/span>1<\/span>)!<\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span><\/span>(<\/span>N<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03be<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>a<\/span>)<\/span><\/span>N<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03be<\/span>\u00a0<\/span>\u4ecb\u4e8e<\/span><\/span>\u00a0<\/span>a<\/span>\u00a0<\/span>\u4e0e<\/span><\/span>\u00a0<\/span>x<\/span>\u00a0<\/span>\u4e4b\u95f4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5b83\u8bf4\u7684\u662f：\u8bef\u5dee\u5927\u5c0f\u53d6\u51b3\u4e8e"\u4e0b\u4e00\u9636\u5bfc\u6570"\u548c"<\/p>\n

(<\/p>\n

x<\/p>\n

\u2212<\/p>\n

a<\/p>\n

)<\/p>\n

N<\/p>\n

+<\/p>\n

1<\/p>\n

(x-a)^{N+1}<\/p>\n

<\/span><\/span>(<\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>a<\/span>)<\/span><\/span>N<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"\u3002\u53ea\u6709\u5f53 <\/p>\n

N<\/p>\n

\u2192<\/p>\n

\u221e<\/p>\n

N\\\\to\\\\infty<\/p>\n

<\/span><\/span>N<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>\u221e<\/span><\/span><\/span><\/span><\/span> \u65f6\u4f59\u9879\u8d8b\u4e8e <\/p>\n

0<\/p>\n

0<\/p>\n

<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>，\u6cf0\u52d2\u7ea7\u6570\u624d\u771f\u6b63\u6536\u655b\u5230 <\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

f(x)<\/p>\n

<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

![\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0](https:\/\/i-blog.csdnimg.cn\/direct\/0dd12a897e92446f8ba36be5d8802eaa.png)<\/p>\n

\u56fe 3 \u7528 <\/p>\n

sin<\/p>\n

\u2061<\/p>\n

x<\/p>\n

\\\\sin x<\/p>\n

<\/span><\/span>sin<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span> \u6f14\u793a：5 \u9636\u90e8\u5206\u548c\u4e0e\u771f\u503c\u4e4b\u95f4\u90a3\u6761\u7c89\u8272\u5e26\u5b50\u5c31\u662f\u4f59\u9879\u3002\u5728\u539f\u70b9\u9644\u8fd1\u5b83\u51e0\u4e4e\u4e3a\u96f6，\u8d8a\u5f80\u4e24\u8fb9\u8d8a\u5bbd\u2014\u2014\u201c\u903c\u8fd1\u5f97\u591a\u597d\u201d，\u5c31\u662f"\u4f59\u9879\u6709\u591a\u5c0f"\u3002<\/p>\n

\n

\u624b\u63a8 <\/p>\n

e<\/p>\n

x<\/p>\n

e^{x}<\/p>\n

<\/span><\/span>e<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>：\u6700\u7701\u4e8b\u7684\u4f8b\u5b50<\/h2>\n

<\/p>\n

e<\/p>\n

x<\/p>\n

e^{x}<\/p>\n

<\/span><\/span>e<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5c55\u5f00\u8d77\u6765\u6700\u8212\u670d，\u56e0\u4e3a\u5b83\u7684\u6bcf\u4e00\u9636\u5bfc\u6570\u90fd\u8fd8\u662f\u81ea\u5df1：<\/p>\n

<\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

f<\/p>\n

\u2032<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

f<\/p>\n

\u2032<\/p>\n

\u2032<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

\u22ef<\/p>\n

=<\/p>\n

e<\/p>\n

x<\/p>\n

f(x)=f'(x)=f''(x)=\\\\cdots=e^{x}<\/p>\n

<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>f<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>f<\/span><\/span>\u2032\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span>=<\/span><\/span><\/span><\/span>e<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5728 <\/p>\n

x<\/p>\n

=<\/p>\n

0<\/p>\n

x=0<\/p>\n

<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span> \u5904，\u6bcf\u4e00\u9636\u5bfc\u6570\u90fd\u7b49\u4e8e <\/p>\n

1<\/p>\n

1<\/p>\n

<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>，\u6240\u4ee5 <\/p>\n

f<\/p>\n

(<\/p>\n

n<\/p>\n

)<\/p>\n

(<\/p>\n

0<\/p>\n

)<\/p>\n

=<\/p>\n

1<\/p>\n

f^{(n)}(0)=1<\/p>\n

<\/span><\/span>f<\/span><\/span>(<\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>：<\/p>\n

<\/p>\n

a<\/p>\n

n<\/p>\n

=<\/p>\n

f<\/p>\n

(<\/p>\n

n<\/p>\n

)<\/p>\n

(<\/p>\n

0<\/p>\n

)<\/p>\n

n<\/p>\n

!<\/p>\n

=<\/p>\n

1<\/p>\n

n<\/p>\n

!<\/p>\n

a_n=\\\\frac{f^{(n)}(0)}{n!}=\\\\frac{1}{n!}<\/p>\n

<\/span><\/span>a<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>!<\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span><\/span>(<\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>0<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>!<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/p>\n

\u56fe 4 \u628a\u7cfb\u6570\u4e00\u4e2a\u4e2a\u5217\u51fa\u6765：<\/p>\n

1<\/p>\n

,<\/p>\n

1<\/p>\n

,<\/p>\n

1<\/p>\n

2<\/p>\n

,<\/p>\n

1<\/p>\n

6<\/p>\n

,<\/p>\n

1<\/p>\n

24<\/p>\n

,<\/p>\n

\u2026<\/p>\n

1,1,\\\\frac12,\\\\frac16,\\\\frac1{24},\\\\dots<\/p>\n

<\/span><\/span>1<\/span>,<\/span><\/span>1<\/span>,<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span><\/span>24<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span><\/span><\/span><\/span>，\u4e8e\u662f<\/p>\n

<\/p>\n

e<\/p>\n

x<\/p>\n

=<\/p>\n

1<\/p>\n

+<\/p>\n

x<\/p>\n

+<\/p>\n

x<\/p>\n

2<\/p>\n

2<\/p>\n

!<\/p>\n

+<\/p>\n

x<\/p>\n

3<\/p>\n

3<\/p>\n

!<\/p>\n

+<\/p>\n

\u22ef<\/p>\n

e^{x}=1+x+\\\\frac{x^{2}}{2!}+\\\\frac{x^{3}}{3!}+\\\\cdots<\/p>\n

<\/span><\/span>e<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span>+<\/span><\/span><\/span><\/span>x<\/span><\/span>+<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>!<\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span><\/span>3<\/span>!<\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u628a <\/p>\n

x<\/p>\n

=<\/p>\n

1<\/p>\n

x=1<\/p>\n

<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u4ee3\u8fdb\u53bb，\u5c31\u56de\u5230\u7b2c 11 \u7bc7\u90a3\u6761 <\/p>\n

e<\/p>\n

=<\/p>\n

\u2211<\/p>\n

1<\/p>\n

\/<\/p>\n

k<\/p>\n

!<\/p>\n

e=\\\\sum 1\/k!<\/p>\n

<\/span><\/span>e<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2211<\/span><\/span>1\/<\/span>k<\/span>!<\/span><\/span><\/span><\/span><\/span>\u2014\u2014\u5b83\u4e0d\u662f\u53e6\u8bf4\u4e00\u5957，\u5c31\u662f <\/p>\n

e<\/p>\n

x<\/p>\n

e^{x}<\/p>\n

<\/span><\/span>e<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u9ea6\u514b\u52b3\u6797\u5c55\u5f00\u5728 <\/p>\n

x<\/p>\n

=<\/p>\n

1<\/p>\n

x=1<\/p>\n

<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u5904\u7684\u53d6\u503c\u3002<\/p>\n

\n

\u518d\u624b\u63a8 <\/p>\n

sin<\/p>\n

\u2061<\/p>\n

x<\/p>\n

\\\\sin x<\/p>\n

<\/span><\/span>sin<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>：\u5bfc\u6570\u5faa\u73af\u51fa\u7b26\u53f7<\/h2>\n

<\/p>\n

sin<\/p>\n

\u2061<\/p>\n

x<\/p>\n

\\\\sin x<\/p>\n

<\/span><\/span>sin<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span> \u7684\u5bfc\u6570\u662f"\u5faa\u73af"\u7684：<\/p>\n

sin<\/p>\n

\u2061<\/p>\n

\u2192<\/p>\n

cos<\/p>\n

\u2061<\/p>\n

\u2192<\/p>\n

\u2212<\/p>\n

sin<\/p>\n

\u2061<\/p>\n

\u2192<\/p>\n

\u2212<\/p>\n

cos<\/p>\n

\u2061<\/p>\n

\u2192<\/p>\n

sin<\/p>\n

\u2061<\/p>\n

\u2192<\/p>\n

\u22ef<\/p>\n

\\\\sin\\\\to\\\\cos\\\\to-\\\\sin\\\\to-\\\\cos\\\\to\\\\sin\\\\to\\\\cdots<\/p>\n

<\/span><\/span>sin<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>cos<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>\u2212<\/span><\/span>sin<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>\u2212<\/span><\/span>cos<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>sin<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span><\/span>\u3002\u5728 <\/p>\n

x<\/p>\n

=<\/p>\n

0<\/p>\n

x=0<\/p>\n

<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span> \u5904\u4f9d\u6b21\u53d6\u503c\u4e3a<\/p>\n

<\/p>\n

sin<\/p>\n

\u2061<\/p>\n

0<\/p>\n

=<\/p>\n

0<\/p>\n

,<\/p>\n

cos<\/p>\n

\u2061<\/p>\n

0<\/p>\n

=<\/p>\n

1<\/p>\n

,<\/p>\n

\u2212<\/p>\n

sin<\/p>\n

\u2061<\/p>\n

0<\/p>\n

=<\/p>\n

0<\/p>\n

,<\/p>\n

\u2212<\/p>\n

cos<\/p>\n

\u2061<\/p>\n

0<\/p>\n

=<\/p>\n

\u2212<\/p>\n

1<\/p>\n

,<\/p>\n

\u2026<\/p>\n

\\\\sin 0=0,\\\\quad \\\\cos 0=1,\\\\quad -\\\\sin 0=0,\\\\quad -\\\\cos 0=-1,\\\\quad \\\\dots<\/p>\n

<\/span><\/span>sin<\/span><\/span>0<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span>,<\/span><\/span><\/span>cos<\/span><\/span>0<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span>,<\/span><\/span><\/span>\u2212<\/span><\/span>sin<\/span><\/span>0<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span>,<\/span><\/span><\/span>\u2212<\/span><\/span>cos<\/span><\/span>0<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span>,<\/span><\/span><\/span>\u2026<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e8e\u662f\u7cfb\u6570\u91cc\u5076\u6570\u9636\u5168\u662f <\/p>\n

0<\/p>\n

0<\/p>\n

<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u3001\u5947\u6570\u9636\u5728 <\/p>\n

0<\/p>\n

0<\/p>\n

<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span> \u4e0e <\/p>\n

\u00b1<\/p>\n

1<\/p>\n

\/<\/p>\n

n<\/p>\n

!<\/p>\n

\\\\pm1\/ n!<\/p>\n

<\/span><\/span>\u00b1<\/span>1\/<\/span>n<\/span>!<\/span><\/span><\/span><\/span><\/span> \u4e4b\u95f4\u4ea4\u66ff：<\/p>\n

<\/p>\n

sin<\/p>\n

\u2061<\/p>\n

x<\/p>\n

=<\/p>\n

x<\/p>\n

\u2212<\/p>\n

x<\/p>\n

3<\/p>\n

3<\/p>\n

!<\/p>\n

+<\/p>\n

x<\/p>\n

5<\/p>\n

5<\/p>\n

!<\/p>\n

\u2212<\/p>\n

x<\/p>\n

7<\/p>\n

7<\/p>\n

!<\/p>\n

+<\/p>\n

\u22ef<\/p>\n

\\\\sin x=x-\\\\frac{x^{3}}{3!}+\\\\frac{x^{5}}{5!}-\\\\frac{x^{7}}{7!}+\\\\cdots<\/p>\n

<\/span><\/span>sin<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>3<\/span>!<\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span><\/span><\/span>5<\/span>!<\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>7<\/span>!<\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>7<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e5f\u5c31\u80fd\u770b\u51fa：\u7cfb\u6570\u91cc\u7684\u6b63\u8d1f\u53f7，\u6765\u81ea"\u5404\u9636\u5bfc\u6570\u5728\u5c55\u5f00\u70b9\u7684\u6b63\u8d1f"，\u4e0d\u662f\u4eba\u4e3a\u89c4\u5b9a\u7684\u3002<\/p>\n

\n

\u524d\u63d0：\u4f59\u9879\u8981\u80fd\u8d8b\u4e8e 0<\/h2>\n

\u6700\u540e\u628a\u903b\u8f91\u8865\u9f50\u3002\u9010\u9636\u5339\u914d\u53ea\u4fdd\u8bc1\u4e86\u7cfb\u6570"\u5e94\u8be5\u662f"\u8fd9\u4e2a\u6837\u5b50；\u8981\u4fdd\u8bc1\u7ea7\u6570\u771f\u7684\u6536\u655b\u5230 <\/p>\n

f<\/p>\n

f<\/p>\n

<\/span><\/span>f<\/span><\/span><\/span><\/span><\/span>，\u9700\u8981 <\/p>\n

R<\/p>\n

N<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

\u2192<\/p>\n

0<\/p>\n

R_N(x)\\\\to0<\/p>\n

<\/span><\/span>R<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u3002\u5bf9 <\/p>\n

e<\/p>\n

x<\/p>\n

,<\/p>\n

sin<\/p>\n

\u2061<\/p>\n

x<\/p>\n

,<\/p>\n

cos<\/p>\n

\u2061<\/p>\n

x<\/p>\n

e^{x},\\\\sin x,\\\\cos x<\/p>\n

<\/span><\/span>e<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>sin<\/span><\/span>x<\/span>,<\/span><\/span>cos<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>，\u53ef\u4ee5\u8bc1\u660e\u5728\u5168\u4f53\u5b9e\u6570\u4e0a <\/p>\n

R<\/p>\n

N<\/p>\n

\u2192<\/p>\n

0<\/p>\n

R_N\\\\to0<\/p>\n

<\/span><\/span>R<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2192<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>（\u5b83\u4eec\u7684\u5404\u9636\u5bfc\u6570\u4e00\u81f4\u6709\u754c，\u800c <\/p>\n

(<\/p>\n

x<\/p>\n

\u2212<\/p>\n

a<\/p>\n

)<\/p>\n

N<\/p>\n

+<\/p>\n

1<\/p>\n

\/<\/p>\n

(<\/p>\n

N<\/p>\n

+<\/p>\n

1<\/p>\n

)<\/p>\n

!<\/p>\n

\u2192<\/p>\n

0<\/p>\n

(x-a)^{N+1}\/(N+1)!\\\\to0<\/p>\n

<\/span><\/span>(<\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>a<\/span>)<\/span><\/span>N<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\/<\/span>(<\/span>N<\/span><\/span>+<\/span><\/span><\/span><\/span>1<\/span>)!<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>），\u6240\u4ee5\u8fd9\u4e9b\u5c55\u5f00\u5904\u5904\u6210\u7acb；\u800c\u5bf9 <\/p>\n

1<\/p>\n

1<\/p>\n

\u2212<\/p>\n

x<\/p>\n

\\\\frac{1}{1-x}<\/p>\n

<\/span><\/span><\/span><\/span>1<\/span>\u2212<\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u4e00\u65e6\u8d8a\u8fc7 <\/p>\n

x<\/p>\n

=<\/p>\n

1<\/p>\n

x=1<\/p>\n

<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>，\u4f59\u9879\u5c31\u4e0d\u518d\u8d8b\u4e8e <\/p>\n

0<\/p>\n

0<\/p>\n

<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u2014\u2014\u8fd9\u5c31\u81ea\u7136\u8fc7\u6e21\u5230\u4e0b\u4e00\u7bc7\u7684\u4e3b\u9898：\u6536\u655b\u534a\u5f84\u3002<\/p>\n

\n

\u5c0f\u7ed3<\/h2>\n

\u6cf0\u52d2\u7cfb\u6570\u7684\u63a8\u5bfc\u53ea\u9700\u4e00\u62db：\u8bbe <\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

\u2211<\/p>\n

a<\/p>\n

n<\/p>\n

x<\/p>\n

n<\/p>\n

f(x)=\\\\sum a_nx^{n}<\/p>\n

<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2211<\/span><\/span>a<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u9010\u9636\u6c42\u5bfc\u5e76\u53d6 <\/p>\n

x<\/p>\n

=<\/p>\n

0<\/p>\n

x=0<\/p>\n

<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u2014\u2014\u6bcf\u6c42\u4e00\u6b21\u5bfc\u5c31"\u66b4\u9732"\u4e00\u4e2a\u7cfb\u6570，\u5f97\u5230 <\/p>\n

f<\/p>\n

(<\/p>\n

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<\/span><\/span>R<\/span><\/span>N<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>（\u62c9\u683c\u6717\u65e5\u4f59\u9879 <\/p>\n

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<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>N<\/span>+<\/span>1<\/span>)!<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span><\/span>(<\/span>N<\/span>+<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03be<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>a<\/span>)<\/span><\/span>N<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>）\u3002<\/p>\n

e<\/p>\n

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e^{x}<\/p>\n

<\/span><\/span>e<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>（\u7cfb\u6570 <\/p>\n

1<\/p>\n

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n<\/p>\n

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1\/n!<\/p>\n

<\/span><\/span>1\/<\/span>n<\/span>!<\/span><\/span><\/span><\/span><\/span>）\u3001<\/p>\n

sin<\/p>\n

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0<\/p>\n

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<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u2014\u2014\u6b63\u662f\u6536\u655b\u534a\u5f84\u8981\u56de\u7b54\u7684\u95ee\u9898\u3002<\/p>\n

\n

OK，\u672c\u7bc7\u5148\u5230\u8fd9\u91cc，\u5982\u6709\u7591\u95ee，\u6b22\u8fce\u8bc4\u8bba\u533a\u7559\u8a00\u8ba8\u8bba，\u795d\u5404\u4f4d\u529f\u529b\u5927\u6da8，\u6280\u672f\u66f4\u4e0a\u4e00\u5c42\u697c！！！<\/p>\n","protected":false},"excerpt":{"rendered":"

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