{"id":2016,"date":"2019-06-18T13:13:13","date_gmt":"2019-06-18T04:13:13","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=2016"},"modified":"2019-11-22T19:41:25","modified_gmt":"2019-11-22T10:41:25","slug":"calculus-e-and-pi","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-e-and-pi\/","title":{"rendered":"\uc790\uc5f0\uc0c1\uc218 \\(e\\)\uc640 \uc6d0\uc8fc\uc728 \\(\\pi\\)\ub294 \ubb34\ub9ac\uc218\uc774\ub2e4"},"content":{"rendered":"<p>\uc790\uc5f0\uc0c1\uc218 \\(e\\)\uc640 \uc6d0\uc8fc\uc728 \\(\\pi\\)\ub294 \\(1,\\) \\(0,\\) \\(i\\)\uc640 \ub354\ubd88\uc5b4 \uc218\ud559\uc5d0\uc11c \uac00\uc7a5 \ub9ce\uc774 \uc0ac\uc6a9\ub418\ub294 \uc0c1\uc218\uc774\ub2e4. \\(\\pi\\)\ub294 \ucd08\ub4f1\ud559\uad50 \uacfc\uc815\uc5d0\uc11c \ucc98\uc74c \ub4f1\uc7a5\ud558\uace0 \\(e\\)\ub294 \uace0\ub4f1\ud559\uad50 \uacfc\uc815\uc5d0\uc11c \ucc98\uc74c \ub4f1\uc7a5\ud558\ub294\ub370, \uc911\ub4f1\ud559\uad50 \uad50\uc721\uacfc\uc815\uc5d0\uc11c \uc774 \ub450 \uc0c1\uc218\uac00 \ubb34\ub9ac\uc218\ub77c\ub294 \uc0ac\uc2e4\uc740 \uc99d\uba85 \uc5c6\uc774 \ubc1b\uc544\ub4e4\uc778\ub2e4.<\/p>\n<p>\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc790\uc5f0\uc0c1\uc218 \\(e\\)\uc640 \uc6d0\uc8fc\uc728 \\(\\pi\\)\uac00 \ubb34\ub9ac\uc218\uc784\uc744 \uc99d\uba85\ud55c\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c \uc18c\uac1c\ud558\ub294 \uc99d\uba85 \ubc29\ubc95\uc740 \uace0\ub4f1\ud559\uad50 \uacfc\uc815\uc758 \ubbf8\uc801\ubd84\uc744 \uacf5\ubd80\ud558\uba74 \uc774\ud574\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"box contentindex\">\n<h3 class=\"indextitle\">\ub0b4\uc6a9 \uc21c\uc11c<\/h3>\n<ul>\n<li><a href=\"#irrationalityofe\">\\(e\\)\uac00 \ubb34\ub9ac\uc218\ub77c\ub294 \uc0ac\uc2e4\uc758 \uc99d\uba85<\/a><\/li>\n<li><a href=\"#irrationalityofpi\">\\(\\pi\\)\uac00 \ubb34\ub9ac\uc218\ub77c\ub294 \uc0ac\uc2e4\uc758 \uc99d\uba85<\/a><\/li>\n<\/ul>\n<h3 class=\"pretitle\">\ubbf8\ub9ac \uc54c\uc544\uc57c \ud560 \ub0b4\uc6a9<\/h3>\n<ul>\n<li>\uc790\uc5f0\uc0c1\uc218 (<a href=\"\/blog\/articles\/calculus-derivatives-of-exponential-and-logarithm-functions\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>\ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834 \ud310\uc815\ubc95 (<a href=\"\/blog\/articles\/calculus-convergence-tests-of-series\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>\ub2e8\uc870\uc218\ub834 \uc815\ub9ac (<a href=\"\/blog\/articles\/calculus-limit-of-a-sequence-introduction\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>\ubd80\ubd84\uc801\ubd84\ubc95 (<a href=\"https:\/\/en.wikipedia.org\/wiki\/Integration_by_parts\">\uad00\ub828 \uae00<\/a>)<\/li>\n<\/ul>\n<\/div>\n<p><a name=\"irrationalityofe\"><\/a><\/p>\n<h3>\\(e\\)\uac00 \ubb34\ub9ac\uc218\ub77c\ub294 \uc0ac\uc2e4\uc758 \uc99d\uba85<\/h3>\n<p>\uc790\uc5f0\uc0c1\uc218\ub97c \uc815\uc758\ud558\ub294 \ubc29\ubc95\uc740 \uc5ec\ub7ec \uac00\uc9c0\uac00 \uc788\uc9c0\ub9cc, \uc77c\ubc18\uacc4 \uace0\ub4f1\ud559\uad50 \uacfc\uc815\uc5d0\uc11c\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 1. (\uc790\uc5f0\uc0c1\uc218)<\/span><\/p>\n<p>\\[e := \\lim_{n\\to\\infty} \\left( 1+ \\frac{1}{n} \\right)^n\\tag{1}\\]\n<\/p><\/div>\n<p>\uba3c\uc800 \uc774\uc640 \uac19\uc740 \uc815\uc758 \ubc29\ubc95\uc774 \ubc14\ub978 \ubc29\ubc95\uc784\uc744 \ubcf4\uc5ec\uc57c \ud55c\ub2e4.<\/p>\n<div class=\"lemma\">\n<p><span class=\"theorem\">\ubcf4\uc870\uc815\ub9ac 1.<\/span><br \/>\n\uc815\uc758 1\uc758 \uadf9\ud55c (1)\uc740 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(e_n := (1+ 1\/n)^n\\)\uc774\ub77c\uace0 \ud558\uc790. \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uc5ec \uc218\uc5f4 \\(\\left\\{e_n \\right\\}\\)\uc774 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uc790.<\/p>\n<p>\uba3c\uc800 \\(\\left\\{ e_n \\right\\}\\)\uc774 \uc99d\uac00\uc218\uc5f4\uc784\uc744 \ubcf4\uc774\uc790. \\(n \\ge 2,\\) \\(x > -1\\)\uc77c \ub54c \ubca0\ub974\ub204\uc774\uc758 \ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[ (1+x)^n > 1+nx .\\]<br \/>\n(\ub9cc\uc57d \uc704 \ubd80\ub4f1\uc2dd\uc744 \ubc30\uc6b0\uc9c0 \uc54a\uc558\ub2e4\uba74 \uc9c1\uc811 \uc99d\uba85\ud574 \ubcf4\uc790. \\(n\\)\uc5d0 \ub300\ud55c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc774\uc6a9\ud558\uba74 \ub41c\ub2e4.)<\/p>\n<p>\uc704 \ubd80\ub4f1\uc2dd\uc5d0 \\(x = -1\/n^2\\)\uc744 \ub300\uc785\ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[ \\left( 1 &#8211; \\frac{1}{n^2} \\right)^n > 1- \\frac{1}{n}\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \uc774 \uc2dd\uc744 \ubcc0\ud615\ud558\uba74<br \/>\n\\[ \\left( 1 + \\frac{1}{n} \\right)^n \\left( 1- \\frac{1}{n} \\right)^n > 1- \\frac{1}{n} \\]<br \/>\n\uc774\uba70, \uc591\ubcc0\uc744 \\( (1- 1\/n)^n\\)\uc73c\ub85c \ub098\ub208 \ub4a4 \ubcc0\ud615\ud558\uba74<br \/>\n\\[\\left( 1+ \\frac{1}{n} \\right)^n > \\left( 1 + \\frac{1}{n-1} \\right)^{n-1}\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \uc989 \\(e_n > e_{n-1}\\)\uc774\ubbc0\ub85c \\(\\left\\{ e_n \\right\\}\\)\uc740 \uc99d\uac00\uc218\uc5f4\uc774\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \\(\\left\\{ e_n \\right\\}\\)\uc774 \uc704\ub85c \uc720\uacc4\uc784\uc744 \ubcf4\uc774\uc790. \\(n\\)\uc774 \ucda9\ubd84\ud788 \ud070 \uc790\uc5f0\uc218\uc77c \ub54c, \uc774\ud56d \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\ne_n &#038;= \\left( 1 + \\frac{1}{n} \\right)^n \\\\[4pt]<br \/>\n&#038;= \\sum_{k=0}^n \\binom{n}{k} \\frac{1}{n^k} \\\\[4pt]<br \/>\n&#038;= 1+1+ \\sum_{k=2}^n \\binom{n}{k} \\frac{1}{n^k} \\\\[4pt]<br \/>\n&#038;= 1+1+ \\sum_{k=2}^n \\frac{n!}{k! (n-k)!} \\frac{1}{n^k} \\\\[4pt]<br \/>\n&#038;= 1+1+ \\sum_{k=2}^n \\frac{1}{k!} \\cdot \\frac{n}{n} \\frac{n-1}{n} \\frac{n-2}{n} \\cdots \\frac{n-k+1}{n} \\tag{2}\\\\[4pt]<br \/>\n&#038;< 1+1+ \\sum_{k=2}^n \\frac{1}{k!} \\tag{3} \\\\[4pt]\n&#038;= 1 + 1 + \\left( \\frac{1}{2!} + \\frac{1}{3!} + \\frac{1}{4!} + \\cdots + \\frac{1}{n!} \\right) \\\\[4pt]\n&#038;< 1 + 1 + \\left( \\frac{1}{2} + \\frac{1}{2^2} + \\frac{1}{2^3} + \\cdots + \\frac{1}{2^{n-1}}\\right) \\\\[4pt]\n&#038;< 1 + 1 + 1 = 3.\\tag{4}\n\\end{align}\\]\n\uc989 \\(\\left\\{ e_n \\right\\}\\)\uc740 \uc704\ub85c \uc720\uacc4\uc774\ub2e4.<\/p>\n<p>\uadf8\ub7ec\ubbc0\ub85c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{ e_n \\right\\}\\)\uc740 \uc218\ub834\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span>\n<\/p>\n<\/div>\n<p>\\(e\\)\uac00 \ubb34\ub9ac\uc218\uc784\uc744 \ubcf4\uc774\ub824\uba74 \ub2e4\uc74c \ub4f1\uc2dd\uc774 \ud544\uc694\ud558\ub2e4.<\/p>\n<div class=\"lemma\">\n<p><span class=\"theorem\">\ubcf4\uc870\uc815\ub9ac 2. (\uc790\uc5f0\uc0c1\uc218\uc758 \ub2e4\ub978 \uc815\uc758)<\/span><\/p>\n<p>\\[\\sum_{n=0}^{\\infty} \\frac{1}{n!} = e\\]\n<\/p><\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(b_n = \\sum_{k=0}^{n} 1\/k!\\)\uc774\ub77c\uace0 \ud558\uace0, \\(\\left\\{ b_n \\right\\}\\)\uc758 \uadf9\ud55c\uc744 \\(B\\)\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\uba85\ubc31\ud788 \\(\\left\\{ b_n \\right\\}\\)\uc740 \uc99d\uac00\uc218\uc5f4\uc774\uace0, (3)\uacfc (4)\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{ b_n \\right\\}\\)\uc740 \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\{ b_n \\}\\)\uc740 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uadf9\ud55c \\(B\\)\ub294 \uc798 \uc815\ub418\uc5c8\ub2e4.<\/p>\n<p>\uc774\uc81c \uc815\ub9ac\uc758 \ub4f1\uc2dd\uc744 \uc99d\uba85\ud558\uc790. \uc989 \\(e=B\\)\uc784\uc744 \uc99d\uba85\ud558\uc790.<\/p>\n<p>\uba3c\uc800 (3)\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\left( 1+\\frac{1}{n} \\right)^n < \\sum_{k=0}^n \\frac{1}{k!}\\]\n\uc774\ub2e4. \uc774 \ubd80\ub4f1\uc2dd\uc758 \uc591\ubcc0\uc5d0 \\(n\\,\\to\\,\\infty\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74 \\(e \\le B\\)\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \\(e \\ge B\\)\uc784\uc744 \ubcf4\uc774\uc790. \uc790\uc5f0\uc218 \\(m\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(n\\)\uc774 \\(m\\)\ubcf4\ub2e4 \ud070 \uc790\uc5f0\uc218\ub77c\uace0 \ud558\uc790. (2)\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\begin{align}<br \/>\n\\left( 1+ \\frac{1}{n} \\right)^n &#038;= 1 + 1 + \\sum_{k=2}^{n} \\frac{1}{k!} \\left( 1- \\frac{1}{n} \\right) \\left( 1- \\frac{2}{n} \\right) \\cdots \\left( 1- \\frac{k-1}{n}\\right) \\\\[4pt]<br \/>\n&#038; > 1+1+ \\sum_{k=2}^{m} \\frac{1}{k!} \\left( 1- \\frac{1}{n} \\right) \\left( 1- \\frac{2}{n} \\right) \\cdots \\left( 1- \\frac{k-1}{n}\\right) \\\\[4pt]<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \ubd80\ub4f1\uc2dd\uc758 \uc591\ubcc0\uc5d0 \\(n\\,\\to\\,\\infty\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74<br \/>\n\\[e \\ge 1 + 1 + \\sum_{k=2}^{m} \\frac{1}{k!} = b_m\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \uc5ec\uae30\uc11c \ub2e4\uc2dc \ubd80\ub4f1\uc2dd\uc758 \uc591\ubcc0\uc5d0 \\(m\\,\\to\\,\\infty\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74<br \/>\n\\(e \\ge B\\)\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<p>\uadf8\ub7ec\ubbc0\ub85c \\(e = B\\)\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc774\uc81c \ube44\ub85c\uc18c \\(e\\)\uac00 \ubb34\ub9ac\uc218\uc784\uc744 \uc99d\uba85\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 3.<\/span><br \/>\n\\(e\\)\ub294 \ubb34\ub9ac\uc218\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(e\\)\uac00 \uc720\ub9ac\uc218\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\(e = p\/q\\)\uc774\uace0 \uc11c\ub85c\uc18c\uc778 \uc790\uc5f0\uc218 \\(p\\)\uc640 \\(q\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<p>\\(b_q = \\sum_{k=0}^{q} 1\/k!\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[\\begin{align}<br \/>\ne-b_q &#038;= \\frac{1}{(q+1)!} + \\frac{1}{(q+2)!} + \\frac{1}{(q+3)!} + \\cdots \\\\[4pt]<br \/>\n&#038;= \\frac{1}{(q+1)!} \\left( 1 + \\frac{1}{q+2} + \\frac{1} {(q+2)(q+3)} + \\cdots \\right) \\\\[4pt]<br \/>\n&#038; < \\frac{1}{(q+1)!} \\left( 1 + \\frac{1}{q+1} + \\frac{1}{(q+1)^2} + \\cdots \\right) \\\\[4pt]\n&#038;= \\frac{1}{(q+1)!} \\frac{1}{1-\\frac{1}{q+1}} \\\\[4pt]\n&#038;= \\frac{1}{(q+1)!} \\frac{q+1}{q} \\\\[4pt]\n&#038;= \\frac{1}{q! q}\n\\end{align}\\]\n\uc774\ubbc0\ub85c\n\\[q! \\left( e-b_q \\right) < \\frac{1}{q} \\le 1\\tag{5}\\]\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(q! e\\)\uc640 \\(q! b_q\\)\uac00 \ubaa8\ub450 \uc790\uc5f0\uc218\uc774\uace0 \\(e > b_q\\)\uc774\ubbc0\ub85c \\(q! \\left( e-b_q \\right)\\) \ub610\ud55c \uc790\uc5f0\uc218\uc774\ub2e4. \uc774\uac83\uc740 (5)\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<\/p>\n<p>\uadf8\ub7ec\ubbc0\ub85c \\(e\\)\ub294 \ubb34\ub9ac\uc218\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span>\n<\/p>\n<\/div>\n<p><a name=\"irrationalityofpi\"><\/a><\/p>\n<h3>\\(\\pi\\)\uac00 \ubb34\ub9ac\uc218\ub77c\ub294 \uc0ac\uc2e4\uc758 \uc99d\uba85<\/h3>\n<p>\\(e\\)\uc640 \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(\\pi\\)\ub97c \uc815\uc758\ud558\ub294 \ubc29\ubc95\ub3c4 \uc5ec\ub7ec \uac00\uc9c0\uac00 \uc788\ub2e4. \ud558\uc9c0\ub9cc \\(\\pi\\)\ub294 \\(e\\)\uc5d0 \ube44\ud574 \uce5c\uc219\ud558\uace0, \ucd08\ub4f1\ud559\uad50\ubd80\ud130 \ud559\ubd80 \uc800\ud559\ub144\uae4c\uc9c0\ub294 \\(\\pi\\)\uc758 \uc815\uc758\ub97c \uc77c\uad00\uc131 \uc788\uac8c \uc0ac\uc6a9\ud558\uae30\uc5d0 \ud63c\ub3d9\ud560 \uc5fc\ub824\uac00 \uc5c6\uc73c\ubbc0\ub85c \uc5ec\uae30\uc11c\ub294 \ub530\ub85c \uc815\uc758\ud558\uc9c0 \uc54a\uace0 \uc2dc\uc791\ud558\uaca0\ub2e4.<\/p>\n<p>\uba3c\uc800 \ub2e4\uc74c \uadf9\ud55c\uc744 \uc99d\uba85\ud558\uc790. \uc774 \uc815\ub9ac\uc758 \ub0b4\uc6a9\uc740 \\(n\\)\uc774 \ucee4\uc9c8 \ub54c \\(a^n\\)\ubcf4\ub2e4 \\(n!\\)\uc774 \ud6e8\uc52c \ube68\ub9ac \ucee4\uc9c4\ub2e4\ub294 \ub73b\uc774\ub2e4.<\/p>\n<div class=\"lemma\">\n<p><span class=\"theorem\">\ubcf4\uc870\uc815\ub9ac 4.<\/span><br \/>\n\\(a\\)\uac00 \uc0c1\uc218\uc77c \ub54c<br \/>\n\\[\\lim_{n\\to\\infty} \\frac{a^n}{n!} = 0.\\tag{6}\\]\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(a=0\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 (6)\uc774 \uc790\uba85\ud558\uac8c \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\\(a > 0\\)\uc778 \uacbd\uc6b0\ub97c \uc99d\uba85\ud558\uc790. \\(m > 2a\\)\uc778 \uc790\uc5f0\uc218 \\(m\\)\uc744 \ud0dd\ud558\uc790. \uadf8\ub7ec\uba74 \\(n > m\\)\uc77c \ub54c<br \/>\n\\[\\begin{align}<br \/>\n\\frac{a^n}{n!}<br \/>\n&#038;= \\frac{a^m}{m!} \\cdot \\frac{a^{n-m}}{(m+1)(m+2) \\cdots n} \\\\[4pt]<br \/>\n&#038;= \\frac{a^m}{m!} \\cdot \\frac{a}{m-1} \\frac{a}{m-2} \\cdots \\frac{a}{n} \\\\[4pt]<br \/>\n&#038;< \\frac{a^m}{m!} \\cdot \\frac{1}{2} \\frac{1}{2} \\cdots \\frac{1}{2} \\\\[4pt]\n&#038;= \\frac{a^m}{m!} \\cdot \\frac{1}{2^{n-m}} \\\\[4pt]\n&#038;= \\frac{a^m}{m!} \\cdot \\frac{1}{2^n \\,2^{-m}} \\\\[4pt]\n&#038;= \\frac{a^m \\,2^m}{m!} \\cdot \\frac{1}{2^n}\n\\end{align}\\]\n\uc989\n\\[0 < \\frac{a^n}{n!} < \\frac{a^m \\,2^m}{m!} \\cdot \\frac{1}{2^n}\\]\n\uc774\ub2e4. \uadf8\ub7f0\ub370\n\\[\\lim_{n\\to\\infty} \\left( \\frac{a^m \\,2^m}{m!} \\cdot \\frac{1}{2^n} \\right) =0\\]\n\uc774\ubbc0\ub85c \uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac(\uc870\uc784 \uc815\ub9ac)\uc5d0 \uc758\ud558\uc5ec\n\\[\\lim_{n\\to\\infty} \\frac{a^n}{n!} = 0\\]\n\uc774\ub2e4.<\/p>\n<p>\\(a < 0\\)\uc77c \ub54c\uc5d0\ub294\n\\[ - \\left\\lvert \\frac{a^n}{n!} \\right\\rvert \\le \\frac{a^n}{n!} \\le \\left\\lvert \\frac{a^n}{n!} \\right\\rvert \\]\n\uc774\uace0, \uc67c\ucabd\uc758 \uc2dd\uacfc \uc624\ub978\ucabd\uc758 \uc2dd\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud568\uc744 \uc774\ubbf8 \uc99d\uba85\ud588\uc73c\ubbc0\ub85c \uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uac00\uc6b4\ub370 \uc2dd\ub3c4 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.\n<span class=\"qed\"><\/span>\n<\/p>\n<\/div>\n<p>\ub2e4\uc74c\uacfc \uac19\uc740 \ubcf4\uc870\uc815\ub9ac\ub3c4 \ud544\uc694\ud558\ub2e4. \uc774 \ubcf4\uc870\uc815\ub9ac\ub294 \ub4a4\uc5d0\uc11c \uc801\ubd84\uc744 \uacc4\uc0b0\ud560 \ub54c \ud544\uc694\ud558\ub2e4.<\/p>\n<div class=\"lemma\">\n<p><span class=\"theorem\">\ubcf4\uc870\uc815\ub9ac 5.<\/span><br \/>\n\\(f\\)\uac00 \uc2e4\uc218 \uc804\uccb4 \uc9d1\ud569 \uc704\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uc774\uace0 \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504\uac00 \uc9c1\uc120 \\(x = p\\)\uc5d0 \ub300\uce6d\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec, \\(2n\\)\uacc4 \ub3c4\ud568\uc218 \\(y = f^{(2n)}(x)\\)\uc758 \uadf8\ub798\ud504\ub3c4 \uc9c1\uc120 \\(x = p\\)\uc5d0 \ub300\uce6d\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(p=0\\)\uc778 \uacbd\uc6b0, \uc989 \\(f\\)\uac00 \uc6b0\ud568\uc218\uc778 \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(p\\ne 0\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \ud568\uc218\uc758 \uadf8\ub798\ud504\ub97c \\(x\\)\ucd95\uc758 \ubc29\ud5a5\uc73c\ub85c \\(-p\\)\ub9cc\ud07c \ud3c9\ud589\uc774\ub3d9\ud558\uc5ec \uc6b0\ud568\uc218\uc758 \uadf8\ub798\ud504\uac00 \ub418\ub3c4\ub85d \ud560 \uc218 \uc788\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<p>\\(f\\)\uac00 \uc6b0\ud568\uc218\uc774\ubbc0\ub85c \uc5f0\uc1c4 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec \\(x > 0\\)\uc778 \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[f &#8216; (-x) = &#8211; f &#8216; (x)\\]<br \/>\n\uc774\ub2e4. \uc591\ubcc0\uc744 \ud55c \ubc88 \ub354 \ubbf8\ubd84\ud558\uba74 \uc5f0\uc1c4 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[- f &#8216; &#8216; (-x) = &#8211; f &#8216; &#8216; (x)\\]<br \/>\n\uc774\ubbc0\ub85c \\(f &#8216; &#8216; (-x) = f &#8216; &#8216; (x)\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f &#8216; &#8216; \\)\uc740 \uc6b0\ud568\uc218\uc774\ub2e4.<\/p>\n<p>\uc774\ub85c\uc368 \uc6b0\ud568\uc218\uc758 \uc774\uacc4\ub3c4\ud568\uc218\uac00 \uc6b0\ud568\uc218\uc784\uc774 \uc99d\uba85\ub418\uc5c8\ub2e4.<\/p>\n<p>\ub450 \ubc88 \ubbf8\ubd84\ud558\ub294 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uba74 \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(f^{(2n)}\\)\uc774 \uc6b0\ud568\uc218\uc784\uc774 \uc99d\uba85\ub41c\ub2e4.<\/p>\n<p>\\(p \\ne 0\\)\uc778 \uacbd\uc6b0 \\(F(x) = f(x+p)\\)\ub77c\uace0 \ud558\uba74 \\(F\\)\ub294 \uc6b0\ud568\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(F^{(2n)}\\)\ub3c4 \uc6b0\ud568\uc218\uc774\ub2e4. \uadf8\ub7f0\ub370<br \/>\n\\[f^{(2n)}(x) = F^{(2n)}(x-p)\\]<br \/>\n\uc774\ubbc0\ub85c \\(y=f^{(2n)}(x)\\)\uc758 \uadf8\ub798\ud504\ub294 \uc9c1\uc120 \\(x=p\\)\uc5d0 \ub300\uce6d\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc99d\uba85\uc744 \uc704\ud574 \ud544\uc694\ud55c \ubcf4\uc870\uc815\ub9ac \uba87 \uac1c\ub97c \ub354 \uc900\ube44\ud558\uc790.<\/p>\n<div class=\"lemma\">\n<p><span class=\"theorem\">\ubcf4\uc870\uc815\ub9ac 6.<\/span><br \/>\n\\(n\\)\uc774 \uc790\uc5f0\uc218\uc774\uace0 \\(P\\)\uac00 \uc815\uc218\uacc4\uc218\ub97c \uac16\ub294 \ub2e4\ud56d\ud568\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(P\\)\uc758 \ucd5c\uc800\ucc28\ud56d\uc758 \ucc28\uc218\uac00 \\(n\\)\uc774\uba74 \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(P^{(k)}(0)\/n!\\)\uc740 \uc815\uc218\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(P\\)\uac00 \\(m\\)\ucc28 \ub2e4\ud56d\ud568\uc218\ub77c\uace0 \ud558\uace0<br \/>\n\\[P(x) = a_m x^m + a_{m-1} x^{m-1} + \\cdots + a_n x^n\\]<br \/>\n\uc73c\ub85c \ub098\ud0c0\ub0b4\uc790.<\/p>\n<p>\\(k < n\\)\uc77c \ub54c \\(k\\)\uacc4 \ub3c4\ud568\uc218 \\(P^(k) (x)\\)\ub294 \uc0c1\uc218\ud56d\uc744 \uac16\uc9c0 \uc54a\ub294 \ub2e4\ud56d\ud568\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(P^{(k)}(0)\/n! = 0\\)\uc774\ub2e4.<\/p>\n<p>\\(k=n\\)\uc77c \ub54c \\(P^{(n)}(x)\\)\uc758 \uc0c1\uc218\ud56d\uc740 \\(n! a_n\\)\uc774\uace0, \ub2e4\ub978 \ud56d\uc740 \ubaa8\ub450 \\(x\\)\ub97c \uc778\uc218\ub85c \uac00\uc9c0\ubbc0\ub85c \\(P^{(n)}(0)\/n! = a_n\\)\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(a_n\\)\uc774 \uc815\uc218\uc774\ubbc0\ub85c \\(P^{(n)}(0)\/n!\\)\ub3c4 \uc815\uc218\uc774\ub2e4.<\/p>\n<p>\ub9c8\ucc2c\uac00\uc9c0\ub85c \\(k > n\\)\uc77c \ub54c \\(P^{(k)}(x)\\)\uc758 \uc0c1\uc218\ud56d\uc740 \uc815\uc218\uacc4\uc218\ub97c \uac16\ub294 \ub2e8\ud56d\uc2dd\uc744 \\(n\\)\ubc88 \uc774\uc0c1 \ubbf8\ubd84\ud558\uc5ec \uc0c1\uc218\uac00 \ub41c \uac83\uc774\ubbc0\ub85c \\(n!\\)\uc5d0 \uc815\uc218\ub97c \uacf1\ud55c \ud615\ud0dc\uc774\ub2e4. \ub530\ub77c\uc11c \\(P^{(k)}(0)\/n!\\)\uc740 \uc815\uc218\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span>\n<\/p>\n<\/div>\n<p>\uc870\uae08 \ub354 \ud798\uc744 \ub0b4\uc790. \uc774 \ubcf4\uc870\uc815\ub9ac\ub294 \ub4a4\uc5d0\uc11c \uc801\ubd84\uc744 \uacc4\uc0b0\ud560 \ub54c \uc2dd\uc774 \uac04\ub2e8\ud574\uc9c0\ub3c4\ub85d \ud558\uae30 \uc704\ud55c \uac83\uc774\ub2e4.<\/p>\n<div class=\"lemma\">\n<p><span class=\"theorem\">\ubcf4\uc870\uc815\ub9ac 7.<\/span><br \/>\n\\(n\\)\uc774 \uc790\uc5f0\uc218\uc774\uace0 \\(f\\)\uac00 \\(2n\\)\ucc28 \ub2e4\ud56d\ud568\uc218\uc774\uba70<br \/>\n\\[F(x) = f(x) &#8211; f &#8221; (x) + f^{(4)} (x) &#8211; f^{(6)}(x) + \\cdots + (-1)^n f^{(2n)}(x)\\tag{7}\\]<br \/>\n\uc774\uba74<br \/>\n\\[\\int_0^\\pi f(x) \\sin x \\, dx = F(\\pi) + F(0) \\tag{8}\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\ubd80\ubd84\uc801\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uc790. \\(\\sin 0 = \\sin \\pi = 0\\)\uc774\ub77c\ub294 \uc0ac\uc2e4\uc744 \uc774\uc6a9\ud558\uc5ec \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\int_0^\\pi f(x) \\sin x \\,dx<br \/>\n&#038;= \\left[ -f (x) \\cos x \\right]_{0}^{\\pi} + \\int_0^\\pi f &#8216; (x) \\cos x \\,dx \\\\[4pt]<br \/>\n&#038;= \\left[ -f (x) \\cos x + f &#8216; (x) \\sin x \\right]_{0}^{\\pi}  &#8211; \\int_0^\\pi f &#8216; &#8216; (x) \\sin x\\,dx \\\\[4pt]<br \/>\n&#038;= \\left[ -f (x) \\cos x \\right]_{0}^{\\pi}  &#8211; \\int_0^\\pi f &#8216; &#8216; (x) \\sin x\\,dx \\\\[4pt]<br \/>\n&#038;= \\left[ -f (x) \\cos x + f &#8216; &#8216; (x) \\cos x \\right]_{0}^{\\pi}  + \\int_0^\\pi f ^{(3)} (x) \\cos x\\,dx \\\\[4pt]<br \/>\n&#038;= \\left[ -f (x) \\cos x + f &#8216; &#8216; (x) \\cos x \\right]_{0}^{\\pi}  &#8211; \\int_0^\\pi f ^{(4)} (x) \\sin x\\,dx \\\\[4pt]<br \/>\n&#038;= \\left[ -f (x) \\cos x + f &#8216; &#8216; (x) \\cos x &#8211; f^{(4)}(x) \\cos x \\right]_{0}^{\\pi}  + \\int_0^\\pi f ^{(5)} (x) \\cos x\\,dx \\\\[4pt]<br \/>\n&#038;\\quad\\quad \\vdots \\\\[4pt]<br \/>\n&#038;= \\left[ -F(x) \\cos x \\right] _0 ^\\pi + (-1)^n \\int_0^{\\pi} f^{(2n+1)}(x) \\sin x \\,dx \\\\[4pt]<br \/>\n&#038;= F(\\pi) + F(0) + (-1)^n \\int_0^\\pi 0 \\,\\sin x\\,dx \\\\[4pt]<br \/>\n&#038;= F(\\pi) + F(0).\\tag*{\\(\\blacksquare\\)}<br \/>\n\\end{align}\\]<\/p>\n<\/div>\n<p>\uc774\uc81c \\(\\pi\\)\uac00 \ubb34\ub9ac\uc218\uc784\uc744 \uc99d\uba85\ud558\uc790.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 8.<\/span><br \/>\n\\(\\pi\\)\ub294 \ubb34\ub9ac\uc218\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\pi\\)\uac00 \uc720\ub9ac\uc218\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\(\\pi = p\/q\\)\uc774\uace0 \uc11c\ub85c\uc18c\uc778 \uc790\uc5f0\uc218 \\(p\\)\uc640 \\(q\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(n\\)\uc774 \uc790\uc5f0\uc218\uc774\uace0<br \/>\n\\[I_n = \\frac{q^n}{n!} \\int_0^\\pi x^n (\\pi -x)^n \\sin x \\,dx\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \ud53c\uc801\ubd84\ud568\uc218 \\( x^n (\\pi -x)^n \\sin x\\)\ub294 \uc801\ubd84 \ubc94\uc704 \ub0b4\uc5d0\uc11c \\(0\\) \uc774\uc0c1\uc758 \uac12\uc744 \uac00\uc9c0\uba70 \\(x = \\pi \/ 2\\)\uc77c \ub54c \ucd5c\ub313\uac12\uc744 \uac00\uc9c0\ubbc0\ub85c<br \/>\n\\[0 < I_n \\le \\frac{q^n}{n!} \\int_0^\\pi \\left( \\frac{\\pi}{2}\\right)^{2n} \\,dx = \\frac{\\pi q^n}{n!} \\left( \\frac{\\pi}{2}\\right)^{2n}\\]\n\uc774\ub2e4. \ubcf4\uc870\uc815\ub9ac 4\uc5d0 \uc758\ud558\uc5ec, \\(n\\,\\to\\,\\infty\\)\uc77c \ub54c\n\\[\\frac{\\pi q^n}{n!} \\left( \\frac{\\pi}{2}\\right)^{2n} \\,\\to\\,0\\]\n\uc774\ubbc0\ub85c, \uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(I_n \\,\\to\\,0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ucda9\ubd84\ud788 \ud070 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec\n\\[0 < I_n  < 1 \\tag{9}\\]\n\uc774\ub2e4. \uc774\uc81c \\(n\\)\uc740 (9)\ub97c \ub9cc\uc871\uc2dc\ud0ac \uc815\ub3c4\ub85c \ucda9\ubd84\ud788 \ud070 \uc790\uc5f0\uc218\ub77c\uace0 \ud558\uace0\n\\[P(x) = x^n \\left( p-qx \\right)^n ,\\,\\, f(x) = \\frac{P(x)}{n!}\\]\n\ub77c\uace0 \ud558\uc790. \\(p\\)\uc640 \\(q\\)\uac00 \uc790\uc5f0\uc218\uc774\ubbc0\ub85c \\(P(x)\\)\ub294 \uc815\uc218\uacc4\uc218\ub97c \uac16\ub294 \\(2n\\)\ucc28 \ub2e4\ud56d\uc2dd\uc774\uba70 \ucd5c\uc800\ucc28\ud56d\uc758 \ucc28\uc218\ub294 \\(n\\)\uc774\ub2e4. \ub610\ud55c \\(\\pi = p\/q\\)\uc774\ubbc0\ub85c\n\\[P(x) = x^n (p-qx)^n = q^n x^n (\\pi - x)^n\\]\n\uc774\uace0\n\\[I_n = \\frac{1}{n!} \\int_0^{\\pi} P(x)\\sin x\\, dx = \\int_0^\\pi f(x) \\sin x\\,dx\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (7)\uc5d0\uc11c \uc815\uc758\ud55c \ud568\uc218 \\(F\\)\ub97c \uc774\uc6a9\ud558\uba74 \ubcf4\uc870\uc815\ub9ac 7\uc5d0 \uc758\ud558\uc5ec \uc704 \uc801\ubd84\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uacc4\uc0b0\ub41c\ub2e4.\n\\[I_n = F(\\pi ) + F(0).\\tag{10}\\]\n\uadf8\ub7f0\ub370 \\(P(x)\\)\ub294 \ucd5c\uc800\ucc28\ud56d\uc758 \ucc28\uc218\uac00 \\(n\\)\uc778 \ub2e4\ud56d\ud568\uc218\uc774\ubbc0\ub85c, \ubcf4\uc870\uc815\ub9ac 6\uc5d0 \uc758\ud558\uc5ec \\(F(0)\\)\uc740 \uc815\uc218\uc758 \ud569\uc774\ub2e4. \uc989 \\(F(0)\\)\uc758 \uac12\uc740 \uc815\uc218\uc774\ub2e4. \ub354\uc6b1\uc774 \ubcf4\uc870\uc815\ub9ac 5\uc5d0 \uc758\ud558\uc5ec \\(F\\)\ub294 \uadf8\ub798\ud504\uac00 \uc9c1\uc120 \\(x = \\pi\/2\\)\uc5d0 \ub300\uce6d\uc778 \ud568\uc218\uc758 \ud569\uc774\ubbc0\ub85c \\(F\\)\uc758 \uadf8\ub798\ud504 \ub610\ud55c \uc9c1\uc120 \\(x = \\pi\/2\\)\uc5d0 \ub300\uce6d\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(F(\\pi) = F(0)\\)\uc774\ub2e4. \uc989 \\(F(\\pi)\\)\uc758 \uac12\ub3c4 \uc815\uc218\uc774\ub2e4.<\/p>\n<p>\uc774\ub85c\uc368 (10)\uc758 \uc6b0\ubcc0\uc740 \uc815\uc218\uc640 \uc815\uc218\uc758 \ud569\uc774\ubbc0\ub85c \\(I_n\\)\uc740 \uc815\uc218\uc774\ub2e4. \uadf8\ub7f0\ub370 (9)\uc5d0 \uc758\ud558\uc5ec \\(I_n\\)\uc740 \\(0\\)\ubcf4\ub2e4 \ud06c\uace0 \\(1\\)\ubcf4\ub2e4 \uc791\ub2e4. \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ub2e4.<\/p>\n<p>\uadf8\ub7ec\ubbc0\ub85c \\(\\pi\\)\ub294 \uc720\ub9ac\uc218\uac00 \uc544\ub2c8\ub2e4.<br \/>\n<span class=\"qed\"><\/span>\n<\/p>\n<\/div>\n<h3>\ucd94\uc2e0<\/h3>\n<p>\uace0\ub4f1\ud559\uad50 \ubbf8\uc801\ubd84 \uc218\uc900\uc5d0\uc11c \uc4f0\ub824\uace0 \ud588\ub294\ub370, \ucc98\uc74c\ubd80\ud130 \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uac00 \ub098\uc640\ubc84\ub838\ub2e4. \ub9dd\ud588\ub098?<\/p>\n<p>\uc544\ub2c8\ub2e4. \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\ub294 \uc26c\uc6b0\ub2c8 \uad00\uc2ec \uc788\uc73c\uba74 \ucc3e\uc544\ubcf4\uae30\ub97c!<\/p>\n<p><!-- --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc790\uc5f0\uc0c1\uc218 \\(e\\)\uc640 \uc6d0\uc8fc\uc728 \\(\\pi\\)\ub294 \\(1,\\) \\(0,\\) \\(i\\)\uc640 \ub354\ubd88\uc5b4 \uc218\ud559\uc5d0\uc11c \uac00\uc7a5 \ub9ce\uc774 \uc0ac\uc6a9\ub418\ub294 \uc0c1\uc218\uc774\ub2e4. \\(\\pi\\)\ub294 \ucd08\ub4f1\ud559\uad50 \uacfc\uc815\uc5d0\uc11c \ucc98\uc74c \ub4f1\uc7a5\ud558\uace0 \\(e\\)\ub294 \uace0\ub4f1\ud559\uad50 \uacfc\uc815\uc5d0\uc11c \ucc98\uc74c \ub4f1\uc7a5\ud558\ub294\ub370, \uc911\ub4f1\ud559\uad50 \uad50\uc721\uacfc\uc815\uc5d0\uc11c \uc774 \ub450 \uc0c1\uc218\uac00 \ubb34\ub9ac\uc218\ub77c\ub294 \uc0ac\uc2e4\uc740 \uc99d\uba85 \uc5c6\uc774 \ubc1b\uc544\ub4e4\uc778\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc790\uc5f0\uc0c1\uc218 \\(e\\)\uc640 \uc6d0\uc8fc\uc728 \\(\\pi\\)\uac00 \ubb34\ub9ac\uc218\uc784\uc744 \uc99d\uba85\ud55c\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c \uc18c\uac1c\ud558\ub294 \uc99d\uba85 \ubc29\ubc95\uc740 \uace0\ub4f1\ud559\uad50 \uacfc\uc815\uc758 \ubbf8\uc801\ubd84\uc744 \uacf5\ubd80\ud558\uba74 \uc774\ud574\ud560 \uc218 \uc788\ub2e4. \ub0b4\uc6a9 \uc21c\uc11c \\(e\\)\uac00 \ubb34\ub9ac\uc218\ub77c\ub294 \uc0ac\uc2e4\uc758 \uc99d\uba85 \\(\\pi\\)\uac00 \ubb34\ub9ac\uc218\ub77c\ub294 \uc0ac\uc2e4\uc758 \uc99d\uba85 \ubbf8\ub9ac&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[321,324,322,319,323,230,325,326,320],"class_list":["post-2016","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-e","tag-324","tag-322","tag-319","tag-323","tag-230","tag-325","tag-326","tag-320"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2016","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=2016"}],"version-history":[{"count":55,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2016\/revisions"}],"predecessor-version":[{"id":4286,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2016\/revisions\/4286"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=2016"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=2016"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=2016"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}