{"id":31157,"date":"2021-03-27T13:00:24","date_gmt":"2021-03-27T13:00:24","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=31157"},"modified":"2025-01-07T03:49:46","modified_gmt":"2025-01-07T03:49:46","slug":"%d8%b5%d9%8a%d8%ba%d8%a9-%d8%b3%d8%aa%d9%8a%d8%b1%d9%84%d9%8a%d9%86%d8%ba","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/ar\/%d8%b5%d9%8a%d8%ba%d8%a9-%d8%b3%d8%aa%d9%8a%d8%b1%d9%84%d9%8a%d9%86%d8%ba\/","title":{"rendered":"\u0635\u064a\u063a\u0629 \u0633\u062a\u064a\u0631\u0644\u064a\u0646\u063a"},"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>\u0635\u064a\u063a\u0629 \u0633\u062a\u064a\u0631\u0644\u064a\u0646\u063a<\/h1>\n<p><em>\u062a\u064f\u0639\u062f \u0635\u064a\u063a\u0629 \u0633\u062a\u064a\u0631\u0644\u064a\u0646\u063a \u0623\u062f\u0627\u0629 \u0623\u0633\u0627\u0633\u064a\u0629 \u0644\u062a\u0628\u0633\u064a\u0637 \u0627\u0644\u062d\u0633\u0627\u0628\u0627\u062a \u0627\u0644\u062e\u0627\u0635\u0629 \u0628\u0645\u0636\u0627\u0639\u0641\u0627\u062a \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0643\u0628\u064a\u0631\u0629\u060c \u062d\u064a\u062b \u062a\u0648\u0641\u0631 \u062a\u0642\u0631\u064a\u0628\u064b\u0627 \u0633\u0631\u064a\u0639\u064b\u0627 \u0648\u0639\u0645\u0644\u064a\u064b\u0627.<\/p>\n<p>\u0647\u0630\u0627 \u0627\u0644\u0646\u0627\u062a\u062c \u0645\u0641\u064a\u062f \u0628\u0634\u0643\u0644 \u062e\u0627\u0635 \u0641\u064a \u0645\u062c\u0627\u0644\u0627\u062a \u0645\u062b\u0644 \u0627\u0644\u062f\u064a\u0646\u0627\u0645\u064a\u0643\u0627 \u0627\u0644\u062d\u0631\u0627\u0631\u064a\u0629\u060c \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u0627\u062a\u060c \u0648\u0627\u0644\u062a\u062d\u0644\u064a\u0644 \u0627\u0644\u062a\u0642\u0627\u0631\u0628\u064a\u060c \u062d\u064a\u062b \u064a\u0643\u0648\u0646 \u0627\u0644\u062a\u0639\u0627\u0645\u0644 \u0645\u0639 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0643\u0628\u064a\u0631\u0629 \u0623\u0645\u0631\u064b\u0627 \u0634\u0627\u0626\u0639\u064b\u0627. \u0641\u0647\u0645 \u062a\u0637\u0648\u064a\u0631\u0647\u0627 \u0644\u0627 \u064a\u064f\u0633\u0647\u0644 \u0627\u0633\u062a\u062e\u062f\u0627\u0645\u0647\u0627 \u0641\u062d\u0633\u0628\u060c \u0628\u0644 \u064a\u064f\u0628\u0631\u0632 \u0623\u064a\u0636\u064b\u0627 \u0623\u0647\u0645\u064a\u062a\u0647\u0627 \u0641\u064a \u0627\u0644\u062d\u0633\u0627\u0628\u0627\u062a \u0627\u0644\u0641\u0639\u0627\u0644\u0629 \u0648\u062d\u0644 \u0627\u0644\u0645\u0634\u0643\u0644\u0627\u062a \u0627\u0644\u0645\u0639\u0642\u062f\u0629.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><strong>\u0623\u0647\u062f\u0627\u0641 \u0627\u0644\u062a\u0639\u0644\u0645:<\/strong><br \/>\n\u0639\u0646\u062f \u0625\u062a\u0645\u0627\u0645 \u0647\u0630\u0647 \u0627\u0644\u062d\u0635\u0629\u060c \u0633\u064a\u0643\u0648\u0646 \u0627\u0644\u0637\u0627\u0644\u0628 \u0642\u0627\u062f\u0631\u064b\u0627 \u0639\u0644\u0649:<\/p>\n<ol>\n<li><strong>\u0641\u0647\u0645<\/strong> \u0627\u0633\u062a\u0646\u062a\u0627\u062c \u0635\u064a\u063a\u0629 \u0633\u062a\u064a\u0631\u0644\u064a\u0646\u063a \u0627\u0646\u0637\u0644\u0627\u0642\u064b\u0627 \u0645\u0646 \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0645\u0636\u0627\u0639\u0641 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u062f\u0627\u0644\u0629 \u063a\u0627\u0645\u0627.<\/li>\n<li><strong>\u062a\u0637\u0628\u064a\u0642<\/strong> \u0635\u064a\u063a\u0629 \u0633\u062a\u064a\u0631\u0644\u064a\u0646\u063a \u0644\u062a\u0642\u0631\u064a\u0628 \u0645\u0636\u0627\u0639\u0641\u0627\u062a \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0643\u0628\u064a\u0631\u0629 \u062c\u062f\u064b\u0627.<\/li>\n<li><strong>\u062d\u0633\u0627\u0628<\/strong> \u0627\u0644\u062a\u0642\u0631\u064a\u0628\u0627\u062a \u0627\u0644\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645\u064a\u0629 \u0644\u0644\u0645\u0636\u0627\u0639\u0641\u0627\u062a \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0623\u062f\u0648\u0627\u062a \u0627\u0644\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645\u0627\u062a \u0648\u0627\u0644\u0623\u0633\u0633 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>\u062c\u062f\u0648\u0644 \u0627\u0644\u0645\u062d\u062a\u0648\u064a\u0627\u062a<\/u>:<\/strong><br \/>\n<a href=\"#1\">\u0627\u0633\u062a\u0646\u062a\u0627\u062c \u0635\u064a\u063a\u0629 \u0633\u062a\u064a\u0631\u0644\u064a\u0646\u063a<\/a><br \/>\n<a href=\"#2\">\u0627\u0644\u062a\u0642\u0631\u064a\u0628 \u0627\u0644\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645\u064a \u0644\u0644\u0645\u0636\u0627\u0639\u0641<\/a><br \/>\n<a href=\"#3\">\u0645\u062b\u0627\u0644: \u062a\u0642\u0631\u064a\u0628 \u0645\u0636\u0627\u0639\u0641 \u0639\u062f\u062f \u0643\u0628\u064a\u0631 \u062c\u062f\u064b\u0627<\/a>\n<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u0627\u0633\u062a\u0646\u062a\u0627\u062c \u0635\u064a\u063a\u0629 \u0633\u062a\u064a\u0631\u0644\u064a\u0646\u063a<\/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>\u064a\u0628\u062f\u0623 \u062a\u0637\u0648\u064a\u0631 \u0635\u064a\u063a\u0629 \u0633\u062a\u064a\u0631\u0644\u064a\u0646\u063a \u0645\u0646 \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0645\u0636\u0627\u0639\u0641 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u062f\u0627\u0644\u0629 \u063a\u0627\u0645\u0627\u060c \u0648\u0627\u0644\u0630\u064a \u064a\u064f\u0639\u0628\u0651\u064e\u0631 \u0639\u0646\u0647 \u0643\u0645\u0627 \u064a\u0644\u064a:<\/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>\u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0647\u0630\u0627 \u0627\u0644\u062a\u0639\u0628\u064a\u0631\u060c \u0646\u0642\u0648\u0645 \u0628\u062a\u063a\u064a\u064a\u0631 \u0627\u0644\u0645\u062a\u063a\u064a\u0631: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t = nx<\/span><\/span>. \u0647\u0630\u0627 \u064a\u0639\u0646\u064a \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x \\in [0, \\infty[<\/span><\/span> \u0648\u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dt = n dx<\/span><\/span>. \u0628\u0639\u062f \u0647\u0630\u0627 \u0627\u0644\u062a\u063a\u064a\u064a\u0631\u060c \u062a\u062a\u062d\u0648\u0644 \u0627\u0644\u062a\u0643\u0627\u0645\u0644\u064a\u0629 \u0643\u0645\u0627 \u064a\u0644\u064a:<\/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>\u0628\u0639\u062f \u0630\u0644\u0643\u060c \u0646\u0642\u0648\u0645 \u0628\u062a\u063a\u064a\u064a\u0631 \u0645\u062a\u063a\u064a\u0631 \u0622\u062e\u0631: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x = 1 + \\dfrac{s}{\\sqrt{n}}<\/span><\/span>. \u0647\u0630\u0627 \u064a\u0639\u0646\u064a:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><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>\u0628\u0639\u062f \u0647\u0630\u0627 \u0627\u0644\u062a\u063a\u064a\u064a\u0631 \u0641\u064a \u0627\u0644\u0645\u062a\u063a\u064a\u0631\u060c \u062a\u0635\u0628\u062d \u0627\u0644\u062a\u0643\u0627\u0645\u0644\u064a\u0629 \u0628\u0627\u0644\u0634\u0643\u0644 \u0627\u0644\u062a\u0627\u0644\u064a:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><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>\u0646\u0633\u062a\u062e\u062f\u0645 \u0627\u0644\u0622\u0646 \u062a\u0648\u0633\u0639\u0629 \u0633\u0644\u0633\u0644\u0629 \u062a\u0627\u064a\u0644\u0648\u0631 \u0644\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645 \u0627\u0644\u0637\u0628\u064a\u0639\u064a:<\/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>\u0639\u0646\u062f \u062a\u0637\u0628\u064a\u0642 \u0647\u0630\u0647 \u0627\u0644\u062a\u0648\u0633\u0639\u0629 \u0639\u0644\u0649 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln\\left(1+\\dfrac{s}{\\sqrt{n}}\\right)<\/span><\/span>\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u062a\u0648\u0633\u064a\u0639 \u0627\u0644\u062a\u0639\u0628\u064a\u0631 \u0627\u0644\u0623\u0633\u064a \u0643\u0627\u0644\u062a\u0627\u0644\u064a:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><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>\u0644\u0630\u0644\u0643\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u0643\u062a\u0627\u0628\u0629 \u0627\u0644\u062a\u0639\u0628\u064a\u0631 \u0627\u0644\u0643\u0627\u0645\u0644 \u0643\u0627\u0644\u062a\u0627\u0644\u064a:<\/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>\u0647\u0630\u0647 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0623\u0633\u0627\u0633\u064a\u0629 \u0644\u062d\u0633\u0627\u0628 \u0645\u0636\u0627\u0639\u0641\u0627\u062a \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0643\u0628\u064a\u0631\u0629 \u062c\u062f\u064b\u0627. \u0639\u0646\u062f\u0645\u0627 \u064a\u0632\u062f\u0627\u062f <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span>\u060c \u062a\u062a\u0644\u0627\u0634\u0649 \u0627\u0644\u062d\u062f\u0648\u062f \u062f\u0627\u062e\u0644 \u0627\u0644\u0645\u062c\u0645\u0648\u0639 \u0641\u064a \u0627\u0644\u0623\u0633\u060c \u062a\u0627\u0631\u0643\u0629 \u0641\u0642\u0637 \u0627\u0644\u062d\u062f \u0627\u0644\u0633\u0627\u0626\u062f. \u064a\u062a\u0645 \u062a\u0628\u0633\u064a\u0637 \u0627\u0644\u062a\u0643\u0627\u0645\u0644\u060c \u0648\u064a\u0645\u0643\u0646 \u062d\u0644\u0647 \u0643\u062a\u0643\u0627\u0645\u0644 \u063a\u0627\u0648\u0633\u064a:<\/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>\u062a\u064f\u0639\u0631\u0641 \u0647\u0630\u0647 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0628\u0627\u0633\u0645 \u0635\u064a\u063a\u0629 \u0633\u062a\u064a\u0631\u0644\u064a\u0646\u063a \u0644\u062d\u0633\u0627\u0628 \u0645\u0636\u0627\u0639\u0641\u0627\u062a \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0643\u0628\u064a\u0631\u0629:<\/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>\u0627\u0644\u062a\u0642\u0631\u064a\u0628 \u0627\u0644\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645\u064a \u0644\u0644\u0645\u0636\u0627\u0639\u0641<\/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>\u0646\u062a\u064a\u062c\u0629 \u0645\u0628\u0627\u0634\u0631\u0629 \u0644\u0635\u064a\u063a\u0629 \u0633\u062a\u064a\u0631\u0644\u064a\u0646\u063a \u0647\u064a \u0627\u0644\u062a\u0642\u0631\u064a\u0628 \u0627\u0644\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645\u064a \u0644\u0644\u0645\u0636\u0627\u0639\u0641. \u0628\u0623\u062e\u0630 \u0627\u0644\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645 \u0627\u0644\u0637\u0628\u064a\u0639\u064a \u0644\u0635\u064a\u063a\u0629 \u0633\u062a\u064a\u0631\u0644\u064a\u0646\u063a\u060c \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><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>\u0641\u064a \u0627\u0644\u062e\u0637\u0648\u0629 \u0627\u0644\u0623\u062e\u064a\u0631\u0629\u060c \u064a\u062a\u0645 \u062a\u0642\u0631\u064a\u0628 \u0625\u0636\u0627\u0641\u064a \u0645\u0646 \u062e\u0644\u0627\u0644 \u0625\u0647\u0645\u0627\u0644 \u0627\u0644\u062d\u062f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{1}{2}\\ln(2n\\pi)<\/span><\/span>. \u0647\u0630\u0627 \u0627\u0644\u062d\u062f \u064a\u0635\u0628\u062d \u063a\u064a\u0631 \u0630\u064a \u0623\u0647\u0645\u064a\u0629 \u0645\u0642\u0627\u0631\u0646\u0629 \u0628\u0640 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\ln(n) - n<\/span><\/span> \u0639\u0646\u062f\u0645\u0627 \u062a\u0643\u0648\u0646 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u0643\u0628\u064a\u0631\u0629.<\/p>\n<p>\u064a\u0645\u0643\u0646 \u0627\u0644\u062a\u062d\u0642\u0642 \u0645\u0646 \u0635\u062d\u0629 \u0647\u0630\u0627 \u0627\u0644\u062a\u0642\u0631\u064a\u0628 \u0628\u062d\u0633\u0627\u0628 \u0627\u0644\u062e\u0637\u0623 \u0627\u0644\u0646\u0633\u0628\u064a \u0628\u064a\u0646 \u0627\u0644\u062a\u0639\u0628\u064a\u0631\u064a\u0646:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rcl}\n\n\\text{\u0627\u0644\u062a\u0642\u0631\u064a\u0628 \u0627\u0644\u0623\u0648\u0644\u064a} &amp; = &amp; \\dfrac{1}{2}\\ln(2n\\pi) + n\\ln(n) - n \\\\ \\\\\n\n\\text{\u0627\u0644\u062a\u0642\u0631\u064a\u0628 \u0627\u0644\u0646\u0647\u0627\u0626\u064a} &amp; = &amp; n\\ln(n) - n \\\\ \\\\\n\n\\text{\u0627\u0644\u062e\u0637\u0623 \u0627\u0644\u0646\u0633\u0628\u064a} &amp;=&amp; \\dfrac{\\text{\u0627\u0644\u062a\u0642\u0631\u064a\u0628 \u0627\u0644\u0646\u0647\u0627\u0626\u064a} - \\text{\u0627\u0644\u062a\u0642\u0631\u064a\u0628 \u0627\u0644\u0623\u0648\u0644\u064a}}{\\text{\u0627\u0644\u062a\u0642\u0631\u064a\u0628 \u0627\u0644\u0623\u0648\u0644\u064a}} \\\\ \\\\\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>\u0646\u062d\u0633\u0628 \u0627\u0644\u0622\u0646 \u0627\u0644\u0646\u0647\u0627\u064a\u0629 \u0639\u0646\u062f\u0645\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n \\to \\infty<\/span><\/span>:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\displaystyle \\lim_{n\\to\\infty} \\text{\u0627\u0644\u062e\u0637\u0623 \u0627\u0644\u0646\u0633\u0628\u064a} &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>\u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a\u060c \u0628\u0645\u0627 \u0623\u0646 \u0627\u0644\u062e\u0637\u0623 \u064a\u0642\u062a\u0631\u0628 \u0645\u0646 \u0627\u0644\u0635\u0641\u0631 \u0639\u0646\u062f\u0645\u0627 \u062a\u0643\u0648\u0646 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u0643\u0628\u064a\u0631\u0629\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0627\u0644\u062a\u0642\u0631\u064a\u0628 \u0627\u0644\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645\u064a \u0627\u0644\u062a\u0627\u0644\u064a \u0628\u062b\u0642\u0629:<\/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>\u0645\u062b\u0627\u0644: \u062a\u0642\u0631\u064a\u0628 \u0645\u0636\u0627\u0639\u0641 \u0639\u062f\u062f \u0643\u0628\u064a\u0631 \u062c\u062f\u064b\u0627<\/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>\u062d\u0633\u0627\u0628 \u0645\u0636\u0627\u0639\u0641 \u0623\u0639\u062f\u0627\u062f \u0643\u0628\u064a\u0631\u0629 \u062c\u062f\u064b\u0627\u060c \u0645\u062b\u0644 <em>10,000!<\/em>\u060c \u064a\u0643\u0627\u062f \u064a\u0643\u0648\u0646 \u0645\u0633\u062a\u062d\u064a\u0644\u064b\u0627 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0627\u0644\u0623\u062f\u0648\u0627\u062a \u0627\u0644\u062a\u0642\u0644\u064a\u062f\u064a\u0629 \u0628\u0633\u0628\u0628 \u0627\u0644\u062d\u062c\u0645 \u0627\u0644\u0643\u0628\u064a\u0631 \u0644\u0644\u0646\u062a\u064a\u062c\u0629. \u0648\u0645\u0639 \u0630\u0644\u0643\u060c \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0627\u0644\u062a\u0642\u0631\u064a\u0628 \u0627\u0644\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645\u064a \u0644\u0644\u0645\u0636\u0627\u0639\u0641 \u0627\u0644\u0645\u0634\u062a\u0642 \u0645\u0646 \u0635\u064a\u063a\u0629 \u0633\u062a\u064a\u0631\u0644\u064a\u0646\u063a\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u062c\u0639\u0644 \u0647\u0630\u0647 \u0627\u0644\u062d\u0633\u0627\u0628\u0627\u062a \u0645\u0645\u0643\u0646\u0629 \u062d\u062a\u0649 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0627\u0644\u0622\u0644\u0627\u062a \u0627\u0644\u062d\u0627\u0633\u0628\u0629 \u0627\u0644\u0628\u0633\u064a\u0637\u0629.<\/p>\n<p>\u062a\u0639\u0637\u064a\u0646\u0627 \u0635\u064a\u063a\u0629 \u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645 \u0627\u0644\u0645\u0636\u0627\u0639\u0641:<\/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>\u0644\u062a\u062d\u0648\u064a\u0644 \u0627\u0644\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645\u0627\u062a \u0627\u0644\u0637\u0628\u064a\u0639\u064a\u0629 (<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln<\/span><\/span>) \u0625\u0644\u0649 \u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645\u0627\u062a \u0630\u0627\u062a \u0627\u0644\u0623\u0633\u0627\u0633 10 (<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\log<\/span><\/span>)\u060c \u0646\u0633\u062a\u062e\u062f\u0645 \u0627\u0644\u0639\u0644\u0627\u0642\u0629:<\/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>\u0648\u0647\u0630\u0627 \u064a\u0639\u0646\u064a:<\/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>\u0644\u0630\u0644\u0643:<\/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>\u0646\u0644\u0627\u062d\u0638 \u0647\u0646\u0627 \u0623\u0646 \u0627\u0644\u062a\u0639\u0628\u064a\u0631 \u0641\u064a \u0627\u0644\u0623\u0633 \u064a\u0635\u0628\u062d \u0642\u0627\u0628\u0644\u0627\u064b \u0644\u0644\u0625\u062f\u0627\u0631\u0629 \u0628\u0627\u0644\u0646\u0633\u0628\u0629 \u0644\u0645\u0639\u0638\u0645 \u0627\u0644\u0622\u0644\u0627\u062a \u0627\u0644\u062d\u0627\u0633\u0628\u0629. \u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a\u060c \u0639\u0644\u0649 \u0627\u0644\u0631\u063a\u0645 \u0645\u0646 \u0623\u0646\u0646\u0627 \u0644\u0627 \u0646\u0633\u062a\u0637\u064a\u0639 \u062a\u0635\u0648\u0631 \u0627\u0644\u0631\u0642\u0645 \u0628\u0633\u0628\u0628 \u062d\u062c\u0645\u0647 \u0627\u0644\u0647\u0627\u0626\u0644\u060c \u0625\u0644\u0627 \u0623\u0646\u0646\u0627 \u0646\u0639\u0644\u0645 \u0623\u0646 \u0627\u0644\u0639\u062f\u062f \u064a\u062a\u0643\u0648\u0646 \u0645\u0646 \u062d\u0648\u0627\u0644\u064a 35,657 \u0631\u0642\u0645\u064b\u0627. \u0647\u0630\u0647 \u0627\u0644\u0637\u0631\u064a\u0642\u0629 \u062a\u062d\u0648\u0644 \u0639\u0645\u0644\u064a\u0629 \u062d\u0633\u0627\u0628\u064a\u0629 \u062a\u0628\u062f\u0648 \u0645\u0633\u062a\u062d\u064a\u0644\u0629 \u0625\u0644\u0649 \u0634\u064a\u0621 \u0645\u0645\u0643\u0646 \u0627\u0644\u062a\u0646\u0641\u064a\u0630.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0635\u064a\u063a\u0629 \u0633\u062a\u064a\u0631\u0644\u064a\u0646\u063a \u062a\u064f\u0639\u062f \u0635\u064a\u063a\u0629 \u0633\u062a\u064a\u0631\u0644\u064a\u0646\u063a \u0623\u062f\u0627\u0629 \u0623\u0633\u0627\u0633\u064a\u0629 \u0644\u062a\u0628\u0633\u064a\u0637 \u0627\u0644\u062d\u0633\u0627\u0628\u0627\u062a \u0627\u0644\u062e\u0627\u0635\u0629 \u0628\u0645\u0636\u0627\u0639\u0641\u0627\u062a \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0643\u0628\u064a\u0631\u0629\u060c \u062d\u064a\u062b \u062a\u0648\u0641\u0631 \u062a\u0642\u0631\u064a\u0628\u064b\u0627 \u0633\u0631\u064a\u0639\u064b\u0627 \u0648\u0639\u0645\u0644\u064a\u064b\u0627. \u0647\u0630\u0627 \u0627\u0644\u0646\u0627\u062a\u062c \u0645\u0641\u064a\u062f \u0628\u0634\u0643\u0644 \u062e\u0627\u0635 \u0641\u064a \u0645\u062c\u0627\u0644\u0627\u062a \u0645\u062b\u0644 \u0627\u0644\u062f\u064a\u0646\u0627\u0645\u064a\u0643\u0627 \u0627\u0644\u062d\u0631\u0627\u0631\u064a\u0629\u060c \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u0627\u062a\u060c \u0648\u0627\u0644\u062a\u062d\u0644\u064a\u0644 \u0627\u0644\u062a\u0642\u0627\u0631\u0628\u064a\u060c \u062d\u064a\u062b \u064a\u0643\u0648\u0646 \u0627\u0644\u062a\u0639\u0627\u0645\u0644 \u0645\u0639 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0643\u0628\u064a\u0631\u0629 \u0623\u0645\u0631\u064b\u0627 \u0634\u0627\u0626\u0639\u064b\u0627. \u0641\u0647\u0645 \u062a\u0637\u0648\u064a\u0631\u0647\u0627 \u0644\u0627 \u064a\u064f\u0633\u0647\u0644 \u0627\u0633\u062a\u062e\u062f\u0627\u0645\u0647\u0627 \u0641\u062d\u0633\u0628\u060c \u0628\u0644 \u064a\u064f\u0628\u0631\u0632 \u0623\u064a\u0636\u064b\u0627 \u0623\u0647\u0645\u064a\u062a\u0647\u0627 \u0641\u064a \u0627\u0644\u062d\u0633\u0627\u0628\u0627\u062a \u0627\u0644\u0641\u0639\u0627\u0644\u0629 \u0648\u062d\u0644 \u0627\u0644\u0645\u0634\u0643\u0644\u0627\u062a \u0627\u0644\u0645\u0639\u0642\u062f\u0629. 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