{"id":31146,"date":"2021-03-27T13:00:20","date_gmt":"2021-03-27T13:00:20","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=31146"},"modified":"2025-01-07T03:33:15","modified_gmt":"2025-01-07T03:33:15","slug":"%d1%84%d0%be%d1%80%d0%bc%d1%83%d0%bb%d0%b0-%d1%81%d1%82%d0%b8%d1%80%d0%bb%d0%b8%d0%bd%d0%b3","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/ru\/%d1%84%d0%be%d1%80%d0%bc%d1%83%d0%bb%d0%b0-%d1%81%d1%82%d0%b8%d1%80%d0%bb%d0%b8%d0%bd%d0%b3\/","title":{"rendered":"\u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433"},"content":{"rendered":"<style>\n\tp, ul, ol {\n\t\ttext-align: justify;\n\t}\n\th1, h2 {\n\t\ttext-align: center;\n\t}\n<\/style>\n<h1>\u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430<\/h1>\n<p><em>\u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430 \u2014 \u044d\u0442\u043e \u0432\u0430\u0436\u043d\u044b\u0439 \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442 \u0434\u043b\u044f \u0443\u043f\u0440\u043e\u0449\u0435\u043d\u0438\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0439 \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0430\u043b\u043e\u0432 \u0431\u043e\u043b\u044c\u0448\u0438\u0445 \u0447\u0438\u0441\u0435\u043b, \u043f\u0440\u0435\u0434\u043b\u0430\u0433\u0430\u044f \u0431\u044b\u0441\u0442\u0440\u043e\u0435 \u0438 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u043d\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435.<\/p>\n<p>\u042d\u0442\u043e\u0442 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u043e\u0441\u043e\u0431\u0435\u043d\u043d\u043e \u043f\u043e\u043b\u0435\u0437\u0435\u043d \u0432 \u0442\u0430\u043a\u0438\u0445 \u043e\u0431\u043b\u0430\u0441\u0442\u044f\u0445, \u043a\u0430\u043a \u0442\u0435\u0440\u043c\u043e\u0434\u0438\u043d\u0430\u043c\u0438\u043a\u0430, \u0442\u0435\u043e\u0440\u0438\u044f \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0435\u0439 \u0438 \u0430\u0441\u0438\u043c\u043f\u0442\u043e\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0430\u043d\u0430\u043b\u0438\u0437, \u0433\u0434\u0435 \u0440\u0430\u0431\u043e\u0442\u0430 \u0441 \u043e\u0447\u0435\u043d\u044c \u0431\u043e\u043b\u044c\u0448\u0438\u043c\u0438 \u0447\u0438\u0441\u043b\u0430\u043c\u0438 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043e\u0431\u044b\u0447\u043d\u044b\u043c \u0434\u0435\u043b\u043e\u043c. \u041f\u043e\u043d\u0438\u043c\u0430\u043d\u0438\u0435 \u0435\u0451 \u0432\u044b\u0432\u043e\u0434\u0430 \u043d\u0435 \u0442\u043e\u043b\u044c\u043a\u043e \u043e\u0431\u043b\u0435\u0433\u0447\u0430\u0435\u0442 \u0435\u0451 \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435, \u043d\u043e \u0438 \u043f\u043e\u0437\u0432\u043e\u043b\u044f\u0435\u0442 \u043e\u0446\u0435\u043d\u0438\u0442\u044c \u0435\u0451 \u0432\u0430\u0436\u043d\u043e\u0441\u0442\u044c \u0432 \u044d\u0444\u0444\u0435\u043a\u0442\u0438\u0432\u043d\u044b\u0445 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f\u0445 \u0438 \u0440\u0435\u0448\u0435\u043d\u0438\u0438 \u0441\u043b\u043e\u0436\u043d\u044b\u0445 \u0437\u0430\u0434\u0430\u0447.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><strong>\u0423\u0447\u0435\u0431\u043d\u044b\u0435 \u0446\u0435\u043b\u0438:<\/strong><br \/>\n\u041f\u043e \u0437\u0430\u0432\u0435\u0440\u0448\u0435\u043d\u0438\u0438 \u044d\u0442\u043e\u0433\u043e \u0443\u0440\u043e\u043a\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442 \u0441\u043c\u043e\u0436\u0435\u0442:<\/p>\n<ol>\n<li><strong>\u041f\u043e\u043d\u044f\u0442\u044c<\/strong> \u0432\u044b\u0432\u043e\u0434 \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430 \u0438\u0437 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0430\u043b\u0430 \u0447\u0435\u0440\u0435\u0437 \u0433\u0430\u043c\u043c\u0430-\u0444\u0443\u043d\u043a\u0446\u0438\u044e.<\/li>\n<li><strong>\u041f\u0440\u0438\u043c\u0435\u043d\u0438\u0442\u044c<\/strong> \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430 \u0434\u043b\u044f \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u044f \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0430\u043b\u043e\u0432 \u043e\u0447\u0435\u043d\u044c \u0431\u043e\u043b\u044c\u0448\u0438\u0445 \u0447\u0438\u0441\u0435\u043b.<\/li>\n<li><strong>\u0412\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c<\/strong> \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u044f \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0430\u043b\u043e\u0432 \u0441 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0445 \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442\u043e\u0432 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u043e\u0432 \u0438 \u0441\u0442\u0435\u043f\u0435\u043d\u0435\u0439.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>\u0421\u041e\u0414\u0415\u0420\u0416\u0410\u041d\u0418\u0415<\/u>:<\/strong><br \/>\n<a href=\"#1\">\u0412\u044b\u0432\u043e\u0434 \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430<\/a><br \/>\n<a href=\"#2\">\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435 \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0430\u043b\u043e\u0432<\/a><br \/>\n<a href=\"#3\">\u041f\u0440\u0438\u043c\u0435\u0440: \u041f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435 \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0430\u043b\u0430 \u043e\u0447\u0435\u043d\u044c \u0431\u043e\u043b\u044c\u0448\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u0430<\/a>\n<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u0412\u044b\u0432\u043e\u0434 \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/CcHCyRR1WrY?si=eTZsj4wBqQ2krELG\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p>\u0412\u044b\u0432\u043e\u0434 \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430 \u043d\u0430\u0447\u0438\u043d\u0430\u0435\u0442\u0441\u044f \u0441 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0430\u043b\u0430 \u0447\u0435\u0440\u0435\u0437 \u0433\u0430\u043c\u043c\u0430-\u0444\u0443\u043d\u043a\u0446\u0438\u044e:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n! =\\Gamma(n+1) = \\displaystyle \\int_0^\\infty t^n e^{-t} \\, dt<\/span><\/span><\/p>\n<p>\u0418\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u044d\u0442\u043e \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435, \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f \u0437\u0430\u043c\u0435\u043d\u0430 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t = nx<\/span><\/span>. \u042d\u0442\u043e \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u0430\u0433\u0430\u0435\u0442, \u0447\u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x \\in [0, \\infty[<\/span><\/span> \u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dt = n dx<\/span><\/span>. \u0421 \u044d\u0442\u043e\u0439 \u0437\u0430\u043c\u0435\u043d\u043e\u0439 \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u0443\u0435\u0442\u0441\u044f \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n! = \\Gamma(n+1) = \\displaystyle \\int_0^\\infty (nx)^n e^{-nx} n \\, dx = n^{n+1} \\int_0^\\infty x^n e^{-nx} dx<\/span><\/span><\/p>\n<p>\u0414\u0430\u043b\u0435\u0435 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u043c \u0432\u0442\u043e\u0440\u0443\u044e \u0437\u0430\u043c\u0435\u043d\u0443 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x = 1 + \\dfrac{s}{\\sqrt{n}}<\/span><\/span>. \u042d\u0442\u043e \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u0430\u0433\u0430\u0435\u0442:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n &amp; s = (x-1)\\sqrt{n}, \\quad s \\in [-\\sqrt{n}, \\infty[ \\\\ \\\\\n\n &amp; dx = \\dfrac{ds}{\\sqrt{n}}\n\n\\end{array}<\/span>\n<p>\u0421 \u044d\u0442\u043e\u0439 \u0437\u0430\u043c\u0435\u043d\u043e\u0439 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439 \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b \u043f\u0440\u0438\u043d\u0438\u043c\u0430\u0435\u0442 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0439 \u0432\u0438\u0434:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\nn! = \\Gamma(n+1) &amp;= \\displaystyle n^{n+1} \\int_{-\\sqrt{n}}^\\infty \\left( 1 + \\dfrac{s}{\\sqrt{n}} \\right)^n e^{-n\\left(1+\\dfrac{s}{\\sqrt{n}}\\right)} \\dfrac{ds}{\\sqrt{n}} \\\\ \\\\\n\n&amp;= \\displaystyle \\dfrac{n^{n+1}}{\\sqrt{n}} \\int_{-\\sqrt{n}}^\\infty e^{n\\ln\\left( 1 + \\dfrac{s}{\\sqrt{n}} \\right)} e^{-n - s\\sqrt{n}} ds \\\\ \\\\\n\n&amp;= \\displaystyle n^n e^{-n} \\sqrt{n} \\int_{-\\sqrt{n}}^\\infty e^{n\\ln\\left(1+\\dfrac{s}{\\sqrt{n}}\\right) - s\\sqrt{n}} ds\n\n\\end{array}\n\n<\/span>\n<p>\u0422\u0435\u043f\u0435\u0440\u044c \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u043c \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0430 \u0432 \u0440\u044f\u0434 \u0422\u0435\u0439\u043b\u043e\u0440\u0430:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln(1+x) = \\displaystyle\\sum_{k=1}^{\\infty} \\dfrac{(-1)^{k+1}x^k}{k} <\/span><\/span><\/p>\n<p>\u041f\u0440\u0438\u043c\u0435\u043d\u044f\u044f \u044d\u0442\u043e \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u043a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln\\left(1+\\dfrac{s}{\\sqrt{n}}\\right)<\/span><\/span>, \u0440\u0430\u0437\u043e\u0432\u044c\u0451\u043c \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u043e\u0435 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\nn\\ln\\left(1+\\dfrac{s}{\\sqrt{n}}\\right) - s\\sqrt{n} &amp; = \\displaystyle n \\left[\\sum_{k=1}^{\\infty} \\dfrac{(-1)^{k+1}\\left(\\dfrac{s}{\\sqrt{n}} \\right)^k}{k} \\right] - s\\sqrt{n} \\\\ \\\\\n\n&amp; = n \\left[ \\dfrac{s}{\\sqrt{n}} - \\dfrac{s^2}{2n} + \\dfrac{s^3}{3n\\sqrt{n}} - \\dfrac{s^4}{4n^2} + \\dfrac{s^5}{5n^2\\sqrt{n}} \\cdots \\right] - s\\sqrt{n} \\\\ \\\\\n\n&amp; = s\\sqrt{n} - \\dfrac{s^2}{2} + \\dfrac{s^3}{3\\sqrt{n}} - \\dfrac{s^4}{4n} + \\dfrac{s^5}{5n\\sqrt{n}} \\cdots - s\\sqrt{n} \\\\ \\\\\n\n&amp; = - \\dfrac{s^2}{2} + \\dfrac{s^3}{3\\sqrt{n}} - \\dfrac{s^4}{4n} + \\dfrac{s^5}{5n\\sqrt{n}} \\cdots \\\\ \\\\\n\n&amp; = - \\dfrac{s^2}{2} + \\displaystyle \\sum_{k=3}^\\infty \\dfrac{(-1)^{k+1}s^k}{k\\sqrt{n^{k-2}}}\n\n\\end{array}\n\n<\/span>\n<p>\u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0437\u0430\u043f\u0438\u0441\u0430\u0442\u044c \u043f\u043e\u043b\u043d\u043e\u0435 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n! = \\Gamma(n+1) = \\displaystyle n^n e^{-n} \\sqrt{n} \\int_{-\\sqrt{n}}^\\infty e^{- \\dfrac{s^2}{2} + \\displaystyle \\sum_{k=3}^\\infty \\dfrac{(-1)^{k+1}s^k}{k\\sqrt{n^{k-2}}}} ds <\/span><\/span><\/p>\n<p>\u042d\u0442\u043e\u0442 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043e\u0441\u043d\u043e\u0432\u043e\u0439 \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0430\u043b\u043e\u0432 \u0431\u043e\u043b\u044c\u0448\u0438\u0445 \u0447\u0438\u0441\u0435\u043b. \u041f\u043e \u043c\u0435\u0440\u0435 \u0443\u0432\u0435\u043b\u0438\u0447\u0435\u043d\u0438\u044f <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span>, \u0441\u043b\u0430\u0433\u0430\u0435\u043c\u044b\u0435 \u0432 \u0441\u0443\u043c\u043c\u0435 \u0432\u043d\u0443\u0442\u0440\u0438 \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0442\u044b \u0441\u0442\u0440\u0435\u043c\u044f\u0442\u0441\u044f \u043a \u043d\u0443\u043b\u044e, \u043e\u0441\u0442\u0430\u0432\u043b\u044f\u044f \u0442\u043e\u043b\u044c\u043a\u043e \u0434\u043e\u043c\u0438\u043d\u0438\u0440\u0443\u044e\u0449\u0438\u0439 \u0447\u043b\u0435\u043d. \u042d\u0442\u043e \u0443\u043f\u0440\u043e\u0449\u0430\u0435\u0442 \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u043c\u043e\u0436\u043d\u043e \u0440\u0435\u0448\u0438\u0442\u044c \u043a\u0430\u043a \u0433\u0430\u0443\u0441\u0441\u043e\u0432 \u0438\u043d\u0442\u0435\u0433\u0440\u0430\u043b:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n! = \\Gamma(n+1) \\approx \\displaystyle n^n e^{-n} \\sqrt{n} \\int_{-\\infty}^\\infty e^{- \\frac{s^2}{2}} ds = n^n e^{-n} \\sqrt{n} \\sqrt{2\\pi} <\/span><\/span><\/p>\n<p><strong>\u042d\u0442\u043e\u0442 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u0438\u0437\u0432\u0435\u0441\u0442\u0435\u043d \u043a\u0430\u043a \u0444\u043e\u0440\u043c\u0443\u043b\u0430 \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430 \u0434\u043b\u044f \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0430\u043b\u043e\u0432 \u0431\u043e\u043b\u044c\u0448\u0438\u0445 \u0447\u0438\u0441\u0435\u043b:<\/strong><\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{n! \\approx \\sqrt{2\\pi n}\\left(\\dfrac{n}{e}\\right)^{n}}<\/span><\/span><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435 \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0430\u043b\u0430<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ASifSl6YgTk?si=8rHCOoUbWoiH962o\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p>\u041f\u0440\u044f\u043c\u044b\u043c \u0441\u043b\u0435\u0434\u0441\u0442\u0432\u0438\u0435\u043c \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435 \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0430\u043b\u0430. \u0412\u0437\u044f\u0432 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043e\u0442 \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430, \u043c\u044b \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rcl}\n\n\\ln(n!) \\approx \\ln\\left( \\sqrt{2n\\pi}\\left(\\dfrac{n}{e}\\right)^{n} \\right) &amp;=&amp; \\dfrac{1}{2}\\ln(2n\\pi) + n\\ln\\left(\\dfrac{n}{e}\\right) \\\\ \\\\\n\n&amp;=&amp;  \\dfrac{1}{2}\\ln(2n\\pi) + n\\ln(n) - n \\\\ \\\\\n\n&amp;\\approx &amp; n\\ln(n) - n\n\n\\end{array}<\/span>\n<p>\u041d\u0430 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0435\u043c \u0448\u0430\u0433\u0435 \u0432\u044b\u043f\u043e\u043b\u043d\u0435\u043d\u043e \u0434\u043e\u043f\u043e\u043b\u043d\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435, \u043f\u0440\u0438 \u043a\u043e\u0442\u043e\u0440\u043e\u043c \u043f\u0440\u0435\u043d\u0435\u0431\u0440\u0435\u0433\u0430\u0435\u0442\u0441\u044f \u0447\u043b\u0435\u043d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{1}{2}\\ln(2n\\pi)<\/span><\/span>. \u042d\u0442\u043e\u0442 \u0447\u043b\u0435\u043d \u0441\u0442\u0430\u043d\u043e\u0432\u0438\u0442\u0441\u044f \u043d\u0435\u0437\u043d\u0430\u0447\u0438\u0442\u0435\u043b\u044c\u043d\u044b\u043c \u043f\u043e \u0441\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044e \u0441 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\ln(n) - n<\/span><\/span> \u0434\u043b\u044f \u0431\u043e\u043b\u044c\u0448\u0438\u0445 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0439 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span>.<\/p>\n<p>\u041e\u0431\u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u0438 \u044d\u0442\u043e\u0433\u043e \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u044f \u0437\u0430\u043a\u043b\u044e\u0447\u0430\u0435\u0442\u0441\u044f \u0432 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0438 \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043e\u0448\u0438\u0431\u043a\u0438 \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f\u043c\u0438:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rcl}\n\n\\text{\u041d\u0430\u0447\u0430\u043b\u044c\u043d\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435} &amp; = &amp; \\dfrac{1}{2}\\ln(2n\\pi) + n\\ln(n) - n \\\\ \\\\\n\n\\text{\u041a\u043e\u043d\u0435\u0447\u043d\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435} &amp; = &amp; n\\ln(n) - n \\\\ \\\\\n\n\\text{\u041e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043e\u0448\u0438\u0431\u043a\u0430} &amp;=&amp; \\dfrac{\\text{\u041a\u043e\u043d\u0435\u0447\u043d\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435} - \\text{\u041d\u0430\u0447\u0430\u043b\u044c\u043d\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435}}{\\text{\u041d\u0430\u0447\u0430\u043b\u044c\u043d\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435}} \\\\ \\\\\n\n&amp;=&amp; \\dfrac{-\\dfrac{1}{2}\\ln(2n\\pi)}{\\dfrac{1}{2}\\ln(2n\\pi) + n\\ln(n) - n}\n\n\\end{array}<\/span>\n<p>\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u043f\u0440\u0435\u0434\u0435\u043b, \u043a\u043e\u0433\u0434\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n \\to \\infty<\/span><\/span>:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\displaystyle \\lim_{n\\to\\infty} \\text{\u041e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043e\u0448\u0438\u0431\u043a\u0430} &amp; = \\displaystyle \\lim_{n\\to\\infty} \\dfrac{-\\dfrac{1}{2}\\ln(2n\\pi)}{\\dfrac{1}{2}\\ln(2n\\pi) + n\\ln(n) - n} \\\\ \\\\\n\n&amp; = \\displaystyle \\lim_{n\\to\\infty} \\dfrac{-\\dfrac{1}{2n}}{\\dfrac{1}{2n} + \\ln(n) + 1 - 1} = 0\n\n\\end{array}<\/span>\n<p>\u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u043e\u0448\u0438\u0431\u043a\u0430 \u0441\u0442\u0440\u0435\u043c\u0438\u0442\u0441\u044f \u043a \u043d\u0443\u043b\u044e \u0434\u043b\u044f \u0431\u043e\u043b\u044c\u0448\u0438\u0445 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0439 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span>, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0441 \u0443\u0432\u0435\u0440\u0435\u043d\u043d\u043e\u0441\u0442\u044c\u044e \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0435 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\ln(n!) \\approx n\\ln(n) - n}<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u041f\u0440\u0438\u043c\u0435\u0440: \u041f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435 \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0430\u043b\u0430 \u043e\u0447\u0435\u043d\u044c \u0431\u043e\u043b\u044c\u0448\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u0430<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/kja9niRWjpg?si=eHkcZYaq0Fgntc3G\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p>\u0412\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0435 \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0430\u043b\u043e\u0432 \u043e\u0447\u0435\u043d\u044c \u0431\u043e\u043b\u044c\u0448\u0438\u0445 \u0447\u0438\u0441\u0435\u043b, \u0442\u0430\u043a\u0438\u0445 \u043a\u0430\u043a <em>10.000!<\/em>, \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u043d\u0435\u0432\u043e\u0437\u043c\u043e\u0436\u043d\u043e \u0441 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u043e\u0431\u044b\u0447\u043d\u044b\u0445 \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442\u043e\u0432 \u0438\u0437-\u0437\u0430 \u0440\u0430\u0437\u043c\u0435\u0440\u0430 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u0430. \u041e\u0434\u043d\u0430\u043a\u043e, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435 \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0430\u043b\u0430, \u0432\u044b\u0432\u0435\u0434\u0435\u043d\u043d\u043e\u0435 \u0438\u0437 \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0441\u0434\u0435\u043b\u0430\u0442\u044c \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u0443\u043f\u0440\u0430\u0432\u043b\u044f\u0435\u043c\u044b\u043c\u0438 \u0434\u0430\u0436\u0435 \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u043a\u0430\u043b\u044c\u043a\u0443\u043b\u044f\u0442\u043e\u0440\u043e\u0432.<\/p>\n<p>\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0444\u043e\u0440\u043c\u0443\u043b\u0430 \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0430\u043b\u0430 \u0432\u044b\u0433\u043b\u044f\u0434\u0438\u0442 \u0442\u0430\u043a:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln(10.000!) \\approx 10.000 \\ln(10.000) - 10.000<\/span><\/span><\/p>\n<p>\u0414\u043b\u044f \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f \u0438\u0437 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u043e\u0432 (<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln<\/span><\/span>) \u0432 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u044b \u043f\u043e \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e 10 (<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\log<\/span><\/span>) \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u0442\u0441\u044f \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0430\u044f \u0441\u0432\u044f\u0437\u044c:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln(10.000!) = \\dfrac{\\log(10.000!)}{\\log(e)}<\/span><\/span><\/p>\n<p>\u042d\u0442\u043e \u043e\u0437\u043d\u0430\u0447\u0430\u0435\u0442, \u0447\u0442\u043e:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\log(10.000!) \\approx \\log(e) \\cdot (10.000 \\ln(10.000) - 10.000)<\/span><\/span><\/p>\n<p>\u0421\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e:<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">10.000! \\approx 10^{\\log(e) \\cdot (10.000 \\ln(10.000) - 10.000)} \\approx 10^{35.657,06}<\/span><\/span><\/p>\n<p>\u041d\u0430 \u044d\u0442\u043e\u043c \u044d\u0442\u0430\u043f\u0435 \u043c\u044b \u0437\u0430\u043c\u0435\u0447\u0430\u0435\u043c, \u0447\u0442\u043e \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u043e\u0435 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u0441\u0442\u0430\u043d\u043e\u0432\u0438\u0442\u0441\u044f \u0443\u043f\u0440\u0430\u0432\u043b\u044f\u0435\u043c\u044b\u043c \u0434\u043b\u044f \u0431\u043e\u043b\u044c\u0448\u0438\u043d\u0441\u0442\u0432\u0430 \u043a\u0430\u043b\u044c\u043a\u0443\u043b\u044f\u0442\u043e\u0440\u043e\u0432. \u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u0445\u043e\u0442\u044f \u043c\u044b \u043d\u0435 \u043c\u043e\u0436\u0435\u043c \u0432\u0438\u0437\u0443\u0430\u043b\u0438\u0437\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u0447\u0438\u0441\u043b\u043e \u0438\u0437-\u0437\u0430 \u0435\u0433\u043e \u043e\u0433\u0440\u043e\u043c\u043d\u043e\u0433\u043e \u0440\u0430\u0437\u043c\u0435\u0440\u0430, \u043c\u044b \u0437\u043d\u0430\u0435\u043c, \u0447\u0442\u043e \u043e\u043d\u043e \u0441\u043e\u0434\u0435\u0440\u0436\u0438\u0442 \u043e\u043a\u043e\u043b\u043e 35.657 \u0446\u0438\u0444\u0440. \u042d\u0442\u043e\u0442 \u043f\u043e\u0434\u0445\u043e\u0434 \u043f\u0440\u0435\u0432\u0440\u0430\u0449\u0430\u0435\u0442, \u043a\u0430\u0437\u0430\u043b\u043e\u0441\u044c \u0431\u044b, \u043d\u0435\u0434\u043e\u0441\u0442\u0438\u0436\u0438\u043c\u044b\u0439 \u0440\u0430\u0441\u0447\u0435\u0442 \u0432 \u0432\u044b\u043f\u043e\u043b\u043d\u0438\u043c\u044b\u0439.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430 \u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u0421\u0442\u0438\u0440\u043b\u0438\u043d\u0433\u0430 \u2014 \u044d\u0442\u043e \u0432\u0430\u0436\u043d\u044b\u0439 \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442 \u0434\u043b\u044f \u0443\u043f\u0440\u043e\u0449\u0435\u043d\u0438\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0439 \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0430\u043b\u043e\u0432 \u0431\u043e\u043b\u044c\u0448\u0438\u0445 \u0447\u0438\u0441\u0435\u043b, \u043f\u0440\u0435\u0434\u043b\u0430\u0433\u0430\u044f \u0431\u044b\u0441\u0442\u0440\u043e\u0435 \u0438 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u043d\u043e\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435. \u042d\u0442\u043e\u0442 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u043e\u0441\u043e\u0431\u0435\u043d\u043d\u043e \u043f\u043e\u043b\u0435\u0437\u0435\u043d \u0432 \u0442\u0430\u043a\u0438\u0445 \u043e\u0431\u043b\u0430\u0441\u0442\u044f\u0445, \u043a\u0430\u043a \u0442\u0435\u0440\u043c\u043e\u0434\u0438\u043d\u0430\u043c\u0438\u043a\u0430, \u0442\u0435\u043e\u0440\u0438\u044f \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0435\u0439 \u0438 \u0430\u0441\u0438\u043c\u043f\u0442\u043e\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0430\u043d\u0430\u043b\u0438\u0437, \u0433\u0434\u0435 \u0440\u0430\u0431\u043e\u0442\u0430 \u0441 \u043e\u0447\u0435\u043d\u044c \u0431\u043e\u043b\u044c\u0448\u0438\u043c\u0438 \u0447\u0438\u0441\u043b\u0430\u043c\u0438 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043e\u0431\u044b\u0447\u043d\u044b\u043c \u0434\u0435\u043b\u043e\u043c. \u041f\u043e\u043d\u0438\u043c\u0430\u043d\u0438\u0435 \u0435\u0451 \u0432\u044b\u0432\u043e\u0434\u0430 \u043d\u0435 \u0442\u043e\u043b\u044c\u043a\u043e \u043e\u0431\u043b\u0435\u0433\u0447\u0430\u0435\u0442 \u0435\u0451 \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435, \u043d\u043e \u0438 \u043f\u043e\u0437\u0432\u043e\u043b\u044f\u0435\u0442 \u043e\u0446\u0435\u043d\u0438\u0442\u044c \u0435\u0451 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":30374,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":41,"footnotes":""},"categories":[680,573,929],"tags":[],"class_list":["post-31146","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-680","category-573","category-929"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - 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