{"id":10375,"date":"2023-07-11T18:14:49","date_gmt":"2023-07-11T10:14:49","guid":{"rendered":"https:\/\/blog.iyatt.com\/?p=10375"},"modified":"2025-07-19T19:50:01","modified_gmt":"2025-07-19T11:50:01","slug":"%e6%b3%b0%e5%8b%92%e5%85%ac%e5%bc%8f","status":"publish","type":"post","link":"https:\/\/blog.iyatt.com\/?p=10375","title":{"rendered":"\u6cf0\u52d2\u5b9a\u7406"},"content":{"rendered":"<p>\u8bbe<code>f(x)<\/code>\u5728\u95ed\u533a\u95f4<code>[a,b]<\/code>\u6709<code>n<\/code>\u9636\u8fde\u7eed\u7684\u5bfc\u6570\uff0c\u5728\u5f00\u533a\u95f4<code>(a,b)<\/code>\u5185\u6709\u76f4\u5230<code>n+1<\/code>\u9636\u5bfc\u6570\uff0c<code class=\"katex-inline\">x_0\\in[a,b]<\/code>\uff0c<code class=\"katex-inline\">x\\in[a,b]<\/code>\u662f\u4efb\u610f\u4e24\u70b9\uff0c\u5219\u81f3\u5c11\u5b58\u5728\u4e00\u4e2a\u70b9\u4ecb\u4e8e<code class=\"katex-inline\">\\xi<\/code>\u4e0e<code>x<\/code>\u4e4b\u95f4\uff0c\u4f7f<code class=\"katex-inline\">f(x)=f(x_0)+f'(x_0)(x-x_0)+\\cdots+\\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+R_n(x)<\/code>,\u79f0 <code class=\"katex-inline\">R_n(x)=\\frac{f^{(n+1)}(\\xi)}{(n+1)!}(x-x_0)^{n+1}<\/code> \u4e3a<strong>\u62c9\u683c\u6717\u65e5\u578b\u4f59\u9879<\/strong>\uff0c\u5373\u4e3a<strong>\u5177\u6709\u62c9\u683c\u6717\u65e5\u4f59\u9879\u7684n\u9636\u6cf0\u52d2\u516c\u5f0f<\/strong>\uff0c\u4f59\u9879\u4e5f\u53ef\u8868\u793a\u4e3a <code class=\"katex-inline\">R_n(x)=o((x-x_0)^n)<\/code>\uff0c\u79f0\u4e3a<strong>\u4f69\u4e9a\u8bfa\u4f59\u9879<\/strong>\uff0c\u5373\u4e3a<strong>\u4f69\u4e9a\u8bfa\u4f59\u9879\u6cf0\u52d2\u516c\u5f0f<\/strong>\u3002\u5f53\u6cf0\u52d2\u516c\u5f0f\u4e2d<code class=\"katex-inline\">x_0=0<\/code>\u65f6\uff0c\u5219\u79f0\u4e3a<strong>\u9ea6\u514b\u52b3\u6797\u516c\u5f0f<\/strong>\u3002<\/p>\n<p>\u51e0\u4e2a\u5e38\u7528\u51fd\u6570\u7684<code>x=0<\/code>\u5904\u5c55\u5f00\u7684\u4f69\u4e9a\u8bfa\u4f59\u9879\u6cf0\u52d2\u516c\u5f0f\u5982\u4e0b\uff1a<br \/>\n\u2460 <code class=\"katex-inline\">e^x=1+x+\\frac{x^2}{2}+\\cdots+\\frac{x^n}{n!}+o(x^n)<\/code><br \/>\n\u2461 <code class=\"katex-inline\">\\sin x=x-\\frac{x^3}{3!}+\\cdots+\\frac{(-1)^n}{(2n+1)!}x^{2n+1}+o(x^{2n+1})<\/code><br \/>\n\u2462 <code class=\"katex-inline\">\\cos x=1-\\frac{x^2}{2}+\\cdots+\\frac{(-1)^n}{(2n)!}x^{2n}+o(x^{2n})<\/code><br \/>\n\u2463 <code class=\"katex-inline\">\\tan x=x+\\frac{x^3}{3}+\\cdots+\\frac{x^{2n+1}}{2n+1}+o(x^{2n+1})<\/code><br \/>\n\u2464 <code class=\"katex-inline\">\\frac{1}{1-x}=1+x+x^2+\\cdots+x^n+o(x^n)<\/code><br \/>\n\u2465 <code class=\"katex-inline\">\\frac{1}{1+x}=1-x+x^2-\\cdots+(-1)^nx^n+o(x^n)<\/code><br \/>\n\u2466 <code class=\"katex-inline\">\\ln(1+x)=x-\\frac{x^2}{2}+\\frac{x^2}{3}-\\cdots+\\frac{(-1)^{n-1}}{n}x^n+o(x^n)<\/code><br \/>\n\u2467 <code class=\"katex-inline\">(1+x)^a=1+ax+\\frac{a(a-1)}{2!}x^2+\\cdots+\\frac{1\\cdot a(a-1)\\cdots(a-n+1)}{n!}x^n+o(x^n)<\/code><br \/>\n\u2468 <code class=\"katex-inline\">\\arctan x=x-\\frac{x^3}{3}+\\frac{x^5}{5}-\\cdots+\\frac{(-1)^n}{2n+1}x^{2n+1}+o(x^{2n+1})<\/code><\/p>\n<p>\u5177\u6709\u62c9\u683c\u6717\u65e5\u4f59\u9879\u76840\u9636\u6cf0\u52d2\u516c\u5f0f\u5c31\u662f\u62c9\u683c\u6717\u65e5\u4e2d\u503c\u516c\u5f0f\uff1b\u5177\u6709\u4f69\u4e9a\u8bfa\u4f59\u9879\u76841\u9636\u6cf0\u52d2\u516c\u5f0f\uff0c\u5c31\u662f\u51fd\u6570\u7684\u5fae\u5206\u548c\u589e\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u5f0f\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u8bbef(x)\u5728\u95ed\u533a\u95f4[a,b]\u6709n\u9636\u8fde\u7eed\u7684\u5bfc\u6570\uff0c\u5728\u5f00\u533a\u95f4(a,b)\u5185\u6709\u76f4\u5230n+1\u9636\u5bfc\u6570\uff0cx_0\\in[a,b]\uff0c [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"zakra_page_container_layout":"customizer","zakra_page_sidebar_layout":"customizer","zakra_remove_content_margin":false,"zakra_sidebar":"customizer","zakra_transparent_header":"customizer","zakra_logo":0,"zakra_main_header_style":"default","zakra_menu_item_color":"","zakra_menu_item_hover_color":"","zakra_menu_item_active_color":"","zakra_menu_active_style":"","zakra_page_header":true,"_lmt_disableupdate":"no","_lmt_disable":"no","footnotes":""},"categories":[1,612],"tags":[],"class_list":["post-10375","post","type-post","status-publish","format-standard","hentry","category-all","category-612"],"modified_by":"IYATT-yx","_links":{"self":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/10375","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=10375"}],"version-history":[{"count":3,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/10375\/revisions"}],"predecessor-version":[{"id":21013,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/10375\/revisions\/21013"}],"wp:attachment":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10375"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10375"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10375"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}