{"id":2018,"date":"2019-06-20T13:14:27","date_gmt":"2019-06-20T04:14:27","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=2018"},"modified":"2025-06-11T17:29:00","modified_gmt":"2025-06-11T08:29:00","slug":"calculus-infinite-series","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-infinite-series\/","title":{"rendered":"\ubb34\ud55c\uae09\uc218"},"content":{"rendered":"<p>\ubb34\ud55c\uae09\uc218\ub294 \uc624\ub798 \uc804\ubd80\ud130 \uc218\ud559\uc790\ub4e4\uc744 \ub2f9\ud639\uc2a4\ub7fd\uac8c \ub9cc\ub4e0 \uc8fc\uc81c \uc911 \ud558\ub098\uc600\ub2e4. \uc608\ucee8\ub300<br \/>\n\\[1 + \\frac{1}{2} + \\frac{1}{2^2} + \\frac{1}{2^3} + \\cdots\\]<br \/>\n\ub294 \uc591\uc218\ub97c \ubb34\ud55c\ud788 \ub9ce\uc774 \ub354\ud568\uc5d0\ub3c4 \ubd88\uad6c\ud558\uace0 \uc218\ub834\ud558\uc9c0\ub9cc<br \/>\n\\[ 1 + \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\cdots\\]<br \/>\n\uc740 \ubc1c\uc0b0\ud55c\ub2e4. <a href=\"https:\/\/en.wikipedia.org\/wiki\/Zeno%27s_paradoxes\">\uc81c\ub17c\uc758 \uc5ed\uc124<\/a>\ub3c4 \uace0\ub300\uc5d0 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ud559\uc790\ub4e4\uc744 \uc5bc\ub9c8\ub098 \uad34\ub86d\ud614\ub294\uc9c0\ub97c \ubcf4\uc5ec\uc8fc\ub294 \ubc29\uc99d\uc774\ub2e4.<\/p>\n<p>\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubb34\ud55c\uae09\uc218\ub97c \uc815\uc758\ud558\uace0 \uc911\uc694\ud55c \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<div class=\"box contentindex\">\n<h3 class=\"indextitle\">\ub0b4\uc6a9 \uc21c\uc11c<\/h3>\n<ul>\n<li><a href=\"#definition\">\ubb34\ud55c\uae09\uc218\uc758 \ub73b<\/a><\/li>\n<li><a href=\"#algebraicproperties\">\uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\uc758 \ub300\uc218\uc801 \uc5f0\uc0b0<\/a><\/li>\n<\/ul>\n<h3 class=\"pretitle\">\ubbf8\ub9ac \uc54c\uc544\uc57c \ud560 \ub0b4\uc6a9<\/h3>\n<ul>\n<li>\uc218\uc5f4\uc758 \uadf9\ud55c (<a href=\"\/blog\/articles\/calculus-limit-of-a-sequence-introduction\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<\/ul>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"definition\"><\/a><\/p>\n<h3>\ubb34\ud55c\uae09\uc218\uc758 \ub73b<\/h3>\n<p>\ubb34\ud55c\uc218\uc5f4<br \/>\n\\[a_1 ,\\,\\, a_2 ,\\,\\, a_3 ,\\,\\, \\cdots\\]<br \/>\n\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uc774 \ubb34\ud55c\uc218\uc5f4\uc758 \ubaa8\ub4e0 \ud56d\uc744 \uc21c\uc11c\ub300\ub85c \ub367\uc148 \uae30\ud638\ub85c \uacb0\ud569\ud558\uc5ec \ub9cc\ub4e0 \uc2dd<br \/>\n\\[a_1 + a_2 + a_3 + \\cdots\\]<br \/>\n\uc744 <span class=\"defined\">\ubb34\ud55c\uae09\uc218<\/span>(infinite series)\ub77c\uace0 \ubd80\ub974\uace0<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n \\tag{1}\\]<br \/>\n\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uadf8\ub9ac\uace0 \\(n\\)\uc774 \uc790\uc5f0\uc218\uc77c \ub54c,<br \/>\n\\[\\sum_{k=1}^{n} a_k \\tag{2}\\]<br \/>\n\ub97c \ubb34\ud55c\uae09\uc218 (1)\uc758 <span class=\"defined\">\ubd80\ubd84\ud569<\/span>(partial sum)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub9cc\uc57d \ubd80\ubd84\ud569\uc758 \uadf9\ud55c<br \/>\n\\[\\lim_{n\\to\\infty} \\sum_{k=1}^{n} a_k\\tag{3}\\]<br \/>\n\uac00 \uc2e4\uc218 \\(S\\)\uc5d0 \uc218\ub834\ud558\uba74 \u2018\ubb34\ud55c\uae09\uc218\uac00 <span class=\"defined\">\uc218\ub834\ud55c\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud558\uace0 \ubd80\ubd84\ud569 \uc218\uc5f4\uc758 \uadf9\ud55c\uac12 \\(S\\)\ub97c \ubb34\ud55c\uae09\uc218 (1)\uc758 <span class=\"defined\">\uac12<\/span>(value) \ub610\ub294 <span class=\"defined\">\ud569<\/span>(sum)\uc774\ub77c\uace0 \ubd80\ub974\uba70<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n = S\\]<br \/>\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ubd80\ubd84\ud569\uc758 \uadf9\ud55c (3)\uc774 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uac70\ub098 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud560 \ub54c\uc5d0\ub3c4 \ub9c8\ucc2c\uac00\uc9c0\ub85c \ub4f1\ud638\ub97c \uc0ac\uc6a9\ud558\uc5ec<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n = \\infty\\]<br \/>\n\ub610\ub294<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n = &#8211; \\infty\\]<br \/>\n\ub85c \ub098\ud0c0\ub0b4\uba70, \uac01\uac01\uc758 \uacbd\uc6b0\ub97c \u2018\ubb34\ud55c\uae09\uc218\uac00 <span class=\"defined\">\uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4<\/span>\u2019, \u2018\ubb34\ud55c\uae09\uc218\uac00 <span class=\"defined\">\uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4<\/span>\u2019\ub77c\uace0 \ud45c\ud604\ud55c\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \ubb34\ud55c\uae09\uc218 \uc218\uc5f4\uc758 \ud56d\uc744 \ub367\uc148\uc73c\ub85c \uacb0\ud569\ud55c \uc2dd\uc744 \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud558\uace0 \ubd80\ubd84\ud569\uc758 \uadf9\ud55c\uc744 \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud558\ub294 \ub450 \uac00\uc9c0 \uc758\ubbf8\ub97c \uac00\uc9c4 \uae30\ud638\uc774\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span><br \/>\n\\(\\lvert r \\rvert < 1\\)\uc77c \ub54c \ubb34\ud55c\uae09\uc218\n\\[\\sum_{n=1}^{\\infty} ar^{n-1}\\tag{4}\\]\n\uc740 \uc218\ub834\ud55c\ub2e4. \uc65c\ub0d0\ud558\uba74\n\\[\\lim_{n\\to\\infty} \\sum_{k=1}^{\\infty} ar^{k-1} = \\lim_{n\\to\\infty} \\frac{a(1-r^n )}{1-r} = \\frac{a}{1-r}\\]\n\ub85c\uc11c \ubd80\ubd84\ud569\uc758 \uadf9\ud55c\uc774 \uc218\ub834\ud558\uae30 \ub54c\ubb38\uc774\ub2e4. \uc774\uc640 \uac19\uc740 \uaf34\uc758 \ubb34\ud55c\uae09\uc218 (4)\ub97c <span class=\"defined\">\ubb34\ud55c\ub4f1\ube44\uae09\uc218<\/span> \ub610\ub294 <span class=\"defined\">\uae30\ud558\uae09\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<br \/>\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2.<\/span><br \/>\n\ubb34\ud55c\uae09\uc218<br \/>\n\\[\\sum_{n=1}^{\\infty} \\frac{1}{n}\\tag{5}\\]<br \/>\n\uc740 \ubc1c\uc0b0\ud55c\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(n=2^p\\)\uc774\uace0 \\(p\\)\uac00 \uc790\uc5f0\uc218\uc77c \ub54c<br \/>\n\\[\\begin{align}<br \/>\n\\sum_{k=1}^{n} \\frac{1}{k} &#038;= \\frac{1}{1} + \\frac{1}{2} + \\frac{1}{3} + \\cdots + \\frac{1}{2^p} \\\\[2pt]<br \/>\n&#038;\\ge \\frac{1}{1} + \\frac{1}{2} + \\left( \\frac{1}{4} + \\frac{1}{4} \\right) + \\left( \\frac{1}{8} + \\frac{1}{8} + \\frac{1}{8} + \\frac{1}{8} \\right) + \\cdots + \\left( \\frac{1}{2^p} + \\frac{1}{2^p} + \\cdots + \\frac{1}{2^p} \\right) \\\\[2pt]<br \/>\n&#038;= \\frac{1}{1} + \\frac{1}{2} + \\frac{1}{4} \\times 2 + \\frac{1}{8} \\times 4 + \\cdots + \\frac{1}{2^p} \\times 2^{p-1} \\\\[2pt]<br \/>\n&#038;= 1+ \\frac{1}{2} + \\frac{1}{2} + \\frac{1}{2} +  \\cdots + \\frac{1}{2} \\\\[2pt]<br \/>\n&#038;= 1+ \\frac{p}{2}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ubbc0\ub85c, \\(n\\to\\infty\\)\uc77c \ub54c \ubd80\ubd84\ud569\uc774 \ubc1c\uc0b0\ud558\uae30 \ub54c\ubb38\uc774\ub2e4. \uc989<br \/>\n\\[\\sum_{n=1}^{\\infty} \\frac{1}{n} = \\infty\\]<br \/>\n\uc774\ub2e4. \uc774\uc640 \uac19\uc740 \uaf34\uc758 \ubb34\ud55c\uae09\uc218 (5)\ub97c <span class=\"defined\">\uc870\ud654\uae09\uc218<\/span>(harmonic series)\ub77c\uace0 \ubd80\ub978\ub2e4.<br \/>\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 3.<\/span><br \/>\n\uc591\uc218\uc778 \ud56d\uacfc \ud640\uc218\uc778 \ud56d\uc774 \ubc88\uac08\uc544\uac00\uba70 \ub098\ud0c0\ub098\ub294 \uc218\uc5f4\uc758 \ubb34\ud55c\uae09\uc218\ub97c <span class=\"defined\">\uad50\ub300\uae09\uc218<\/span>(alternating series)\ub77c\uace0 \ubd80\ub978\ub2e4. \uad50\ub300\uae09\uc218<br \/>\n\\[\\sum_{n=1}^{\\infty} (-1)^n \\]<br \/>\n\uc740 \ubc1c\uc0b0\ud55c\ub2e4. \uc65c\ub0d0\ud558\uba74 \ubd80\ubd84\ud569 \uc218\uc5f4<br \/>\n\\[\\sum_{k=1}^{n} (-1)^k\\]<br \/>\n\uc758 \uac12\uc740, \\(n\\)\uc774 \uc9dd\uc218\uc77c \ub54c\uc5d0\ub294 \\(0\\)\uc774\uace0, \\(n\\)\uc774 \ud640\uc218\uc77c \ub54c\uc5d0\ub294 \\(-1\\)\uc774\ubbc0\ub85c, \\(n\\to\\infty\\)\uc77c \ub54c \ubd80\ubd84\ud569\uc774 \uc9c4\ub3d9\ud558\uae30 \ub54c\ubb38\uc774\ub2e4.<br \/>\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<p>\uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc911\uc694\ud55c \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span><br \/>\n\ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_n \\)\uc774 \uc218\ub834\ud558\uba74 \\(a_n \\to 0\\)\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\uc758 \ud569\uc744 \\(S\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(n\\ge 2\\)\uc77c \ub54c<br \/>\n\\[a_n = \\sum_{k=1}^{n} a_k &#8211; \\sum_{k=1}^{n-1} a_k\\]<br \/>\n\uc774\uba70, \ub4f1\uc2dd\uc758 \uc591\ubcc0\uc5d0 \\(n\\to\\infty\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74<br \/>\n\\[\\lim_{n\\to\\infty} a_n = S-S =0\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.<\/span><br \/>\n\\(a \\ne 0\\)\uc774\uace0 \\(\\lvert r \\rvert \\ge 1\\)\uc77c \ub54c \uae30\ud558\uae09\uc218 (4)\ub294 \ubc1c\uc0b0\ud55c\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(n\\to\\infty\\)\uc77c \ub54c \\(ar^{n-1} \\nrightarrow 0\\)\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.<br \/>\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<div class=\"remark\">\n<p><span class=\"remark\">\ucc38\uace0.<\/span><br \/>\n\uc815\ub9ac 1\uc758 \uc5ed\uc740 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc608\ub97c \ub4e4\uba74, \uc870\ud654\uae09\uc218 (5)\ub294 \uc77c\ubc18\ud56d\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0\ub9cc \ubb34\ud55c\uae09\uc218\ub294 \ubc1c\uc0b0\ud55c\ub2e4.<br \/>\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<p>\ubb34\ud55c\uae09\uc218\uc5d0\uc11c \uc720\ud55c \uac1c\uc758 \ud56d\uc758 \uac12\uc744 \ubc14\uafb8\uac70\ub098 \uc81c\uac70\ud558\uac70\ub098 \ucd94\uac00\ud574\ub3c4 \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834\uc131\uc740 \ubcc0\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc608\ucee8\ub300 \\(p\\)\uac00 \uc790\uc5f0\uc218\uc77c \ub54c, \uace0\uc815\ub41c \uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \ubb34\ud55c\uae09\uc218<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n\\tag{6}\\]<br \/>\n\uc774 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740<br \/>\n\\[\\sum_{n=p}^{\\infty} a_n\\]<br \/>\n\uc774 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4. \uc989 \ubb34\ud55c\uae09\uc218\uc758 \uac12\uc5d0 \uad00\uc2ec\uc774 \uc5c6\uace0 \uc218\ub834\uacfc \ubc1c\uc0b0 \uc5ec\ubd80\uc5d0\ub9cc \uad00\uc2ec\uc744 \uac00\uc9c8 \ub54c\uc5d0\ub294 \ubb34\ud55c\uae09\uc218\uac00 \uc2dc\uc791\ud558\ub294 \ud56d\uc758 \ucca8\uc790\ub294 \uc0c1\uad00 \uc5c6\ub2e4\ub294 \uac83\uc774\ub2e4. \uc774\uc640 \uac19\uc740 \uc0ac\uc2e4 \ub54c\ubb38\uc5d0 (6)\uacfc \uac19\uc740 \ubb34\ud55c\uae09\uc218\ub97c \uac04\ub2e8\ud558\uac8c<br \/>\n\\[\\sum a_n \\]<br \/>\n\uc73c\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"algebraicproperties\"><\/a><\/p>\n<h3>\uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\uc758 \ub300\uc218\uc801 \uc5f0\uc0b0<\/h3>\n<p>\ub367\uc148, \ube84\uc148, \uacf1\uc148\uacfc \uad00\ub828\ub41c \ubb34\ud55c\uae09\uc218\uc758 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<p>\ub450 \ubb34\ud55c\uae09\uc218<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n ,\\quad \\sum_{n=1}^{\\infty} b_n\\]<br \/>\n\uc774 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\begin{align}<br \/>\n\\sum_{n=1}^{\\infty} \\left( a_n \\pm b_n \\right)<br \/>\n&#038;= \\lim_{n\\to\\infty} \\sum_{k=1}^{n} \\left( a_n \\pm b_n \\right) \\\\[2pt]<br \/>\n&#038;= \\lim_{n\\to\\infty} \\left( \\sum_{k=1}^{n} a_n \\pm \\sum_{k=1}^{n} b_n \\right) \\\\[2pt]<br \/>\n&#038;= \\lim_{n\\to\\infty} \\sum_{k=1}^{n} a_n \\pm \\lim_{n\\to\\infty} \\sum_{k=1}^n b_n \\\\[2pt]<br \/>\n&#038;= \\sum_{n=1}^{\\infty} a_n \\pm \\sum_{n=1}^{\\infty} b_n<br \/>\n\\end{align}\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. (\ub2e8, \uc704 \uc2dd\uc5d0\uc11c \ubcf5\ubd80\ud638\ub294 \ubaa8\ub450 \ub3d9\uc21c\uc774\ub2e4.) \ub610\ud55c, \uac19\uc740 \ubc29\ubc95\uc73c\ub85c, \\(k\\)\uac00 \uc0c1\uc218\uc77c \ub54c<br \/>\n\\[\\sum_{n=1}^{\\infty} ka_n = k\\sum_{n=1}^{\\infty}a_n\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<p>\ub450 \ubb34\ud55c\uae09\uc218\uc758 \uacf1\uc740 \uc870\uae08 \ub354 \ubcf5\uc7a1\ud558\ub2e4. \uacf1\uc744 \ub17c\ud558\uae30 \uc704\ud574 \uba3c\uc800 \uc808\ub300\uc218\ub834\uc758 \uac1c\ub150\uc744 \ub3c4\uc785\ud558\uc790. \ubb34\ud55c\uae09\uc218<br \/>\n\\[\\sum_{n=1}^{\\infty} \\left\\lvert a_n \\right\\rvert\\]<br \/>\n\uc774 \uc218\ub834\ud560 \ub54c \u2018\\(\\sum a_n \\)\uc740 <span class=\"defined\">\uc808\ub300\uc218\ub834<\/span>(absolutely converge)\ud55c\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uadf8\ub9ac\uace0 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud558\uc9c0\ub9cc \uc808\ub300\uc218\ub834\ud558\uc9c0 \uc54a\ub294 \uacbd\uc6b0, \u2018\ubb34\ud55c\uae09\uc218\ub294 <span class=\"defined\">\uc870\uac74\uc218\ub834<\/span>(conditionally converge)\ud55c\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 2.<\/span><br \/>\n\uc808\ub300\uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\ub294 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\n\\(\\sum a_n \\)\uc774 \uc808\ub300\uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub450 \ubb34\ud55c\uae09\uc218<br \/>\n\\[\\sum \\left\\lvert a_n \\right\\rvert ,\\quad \\sum 2 \\left\\lvert a_n \\right\\rvert\\]<br \/>\n\uc740 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7f0\ub370<br \/>\n\\[0\\le a_n + \\left\\lvert a_n \\right\\rvert \\le 2\\left\\lvert a_n \\right\\rvert\\]<br \/>\n\uc774\ubbc0\ub85c \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\sum_{k=1}^{n} \\left( a_n + \\left\\lvert a_n \\right\\rvert \\right) \\le 2\\sum_{n=1}^{\\infty} \\left\\lvert a_n \\right\\rvert\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(\\sum_{k=1}^{n} \\left( a_k + \\left\\lvert a_k \\right\\rvert \\right)\\)\ub294<br \/>\n\\(n\\)\uc744 \ubcc0\uc218\ub85c \ud558\ub294 \uc99d\uac00\uc218\uc5f4\uc774\ubbc0\ub85c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\sum_{n=1}^{\\infty} \\left( a_n + \\left\\lvert a_n \\right\\rvert \\right) = \\lim_{n\\to\\infty}\\sum_{k=1}^{n} \\left( a_k + \\left\\lvert a_k \\right\\rvert \\right) \\]<br \/>\n\ub294 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\sum a_n\\)\uc740<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n = \\sum_{n=1}^{\\infty} \\left[ \\left( a_n + \\left\\lvert a_n \\right\\rvert \\right) &#8211; \\left\\lvert a_n \\right\\rvert \\right]\\]<br \/>\n\ub85c\uc11c \uc218\ub834\ud558\ub294 \ub450 \ubb34\ud55c\uae09\uc218\uc758 \ud569\uc774\ubbc0\ub85c \uc218\ub834\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"remark\">\n<p><span class=\"remark\">\ucc38\uace0.<\/span><br \/>\n\uc815\ub9ac 2\uc758 \uc5ed\uc740 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc989 \uc218\ub834\ud558\ub294 \uae09\uc218\uac00 \ubaa8\ub450 \uc808\ub300\uc218\ub834\ud558\uc9c0\ub294 \uc54a\ub294\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \ubb34\ud55c\uae09\uc218<br \/>\n\\[\\sum_{n=1}^{\\infty} \\frac{(-1)^n}{n}\\tag{7}\\]<br \/>\n\uc740 <a href=\"https:\/\/en.wikipedia.org\/wiki\/Alternating_series_test\">\uad50\ub300\uae09\uc218 \ud310\uc815\ubc95<\/a>\uc5d0 \uc758\ud558\uc5ec \uc218\ub834\ud558\uc9c0\ub9cc, \uc774 \ubb34\ud55c\uae09\uc218\uc758 \uac01 \ud56d\uc5d0 \uc808\ub313\uac12\uc744 \uc50c\uc6b4 \ubb34\ud55c\uae09\uc218\ub294 (5)\uc758 \uc870\ud654\uae09\uc218\uc774\ubbc0\ub85c \ubc1c\uc0b0\ud55c\ub2e4. \uc774\uc640 \uac19\uc740 \uae09\uc218 (7)\uc744 <span class=\"defined\">\uad50\ub300\uc870\ud654\uae09\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<br \/>\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<p>\uc774\uc81c \ubb34\ud55c\uae09\uc218\uc758 \uacf1\uc744 \ud558\ub098\uc758 \ubb34\ud55c\uae09\uc218\ub85c \ub098\ud0c0\ub0b4\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 3. (\ubb34\ud55c\uae09\uc218\uc5d0 \ub300\ud55c <a href=\"https:\/\/en.wikipedia.org\/wiki\/Franz_Mertens\">Mertens<\/a>\uc758 \uc815\ub9ac)<\/span><\/p>\n<p>\ub450 \ubb34\ud55c\uae09\uc218<br \/>\n\\[\\sum_{n=0}^{\\infty} a_n ,\\quad \\sum_{n=0}^{\\infty} b_n \\]<br \/>\n\uc911 \ud558\ub098 \uc774\uc0c1\uc774 \uc808\ub300\uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0<br \/>\n\\[c_n = \\sum_{k=0}^{n} a_k\\,b_{n-k}\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=0}^{\\infty} c_n \\)\uc740 \uc218\ub834\ud558\uace0<br \/>\n\\[ \\left(\\sum_{n=0}^{\\infty} a_n \\right) \\left( \\sum_{n=0}^{\\infty} b_n \\right) = \\sum_{n=0}^{\\infty} c_n\\tag{8}\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc774\ub54c \\(\\sum c_n\\)\uc744 \\(\\sum a_n\\)\uacfc \\(\\sum b_n\\)\uc758 <span class=\"defined\">\ucf54\uc2dc \uacf1<\/span>(Cauchy product)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>(\ubbf8\uc801\ubd84\ud559\uc744 \ucc98\uc74c \uacf5\ubd80\ud558\ub294 \uc0ac\ub78c\uc740 \uc774 \uc99d\uba85\uc744 \uc0dd\ub7b5\ud574\ub3c4 \uad1c\ucc2e\ub2e4.)<\/p>\n<p>\\(\\sum a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud558\ub294 \uacbd\uc6b0\ub9cc \uc99d\uba85\ud558\uba74 \ub41c\ub2e4. \ub9cc\uc57d \\(\\sum b_n\\)\uc774 \uc808\ub300\uc218\ub834\ud55c\ub2e4\uba74 \ub450 \ubb34\ud55c\uae09\uc218\uc758 \uc5ed\ud560\uc744 \ubc14\uafb8\uba74 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<p>\ud45c\uae30\ub97c \uac04\ub2e8\ud558\uac8c \ud558\uae30 \uc704\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.<br \/>\n\\[\\begin{align}<br \/>\nA_n &#038;= \\sum_{k=0}^{n} a_k , \\quad A = \\sum_{n=0}^{\\infty} a_n, \\\\[2pt]<br \/>\nB_n &#038;= \\sum_{k=0}^{n} b_k , \\quad B = \\sum_{n=0}^{\\infty} b_n, \\\\[2pt]<br \/>\nC_n &#038;= \\sum_{k=0}^{n} c_k , \\quad \\beta_n = B_n &#8211; B.<br \/>\n\\end{align}\\]<br \/>\n\uadf8\ub7ec\uba74 \ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nC_n &#038;= a_0 b_0 + \\left( a_0 b_1 + a_1 b_0 \\right) + \\cdots + \\left( a_0 b_n + a_1 b_{n-1} + \\cdots + a_n b_0 \\right) \\\\[6pt]<br \/>\n&#038;= a_0 B_n + a_1 B_{n-1} + \\cdots + a_n B_0 \\\\[6pt]<br \/>\n&#038;= a_0 (B+\\beta_n ) + a_1 (B+\\beta_{n-1}) + \\cdots + a_n (B+\\beta_0 ) \\\\[6pt]<br \/>\n&#038;= A_n B + a_0 \\beta_n + a_1 \\beta_{n-1} + \\cdots + a_n \\beta_0 .<br \/>\n\\end{align}\\]<br \/>\n\uc5ec\uae30\uc11c<br \/>\n\\[\\gamma_n = a_0 \\beta_n + a_1 \\beta_{n-1} + \\cdots + a_n \\beta_0\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uc790. \\(A_n B \\to AB\\)\uc774\ubbc0\ub85c, \\(\\gamma_n \\to 0\\)\uc784\uc744 \ubcf4\uc774\uae30\ub9cc \ud558\uba74 \\(C_n \\to AB\\)\uac00 \uc99d\uba85\ub41c\ub2e4.<br \/>\n\\[\\alpha = \\sum_{n=0}^{\\infty} \\left\\lvert a_n \\right\\rvert\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\epsilon > 0\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<\/p>\n<p>\\(\\sum b_n\\)\uc774 \\(B\\)\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c \uc790\uc5f0\uc218 \\(N_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n\\ge N_1\\)\uc77c \ub54c<br \/>\n\\[\\left\\lvert \\beta_n \\right\\rvert < \\frac{\\epsilon}{2\\alpha +1}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.\n\\[K = \\max \\left\\{ \\left\\lvert \\beta_0 \\right\\rvert ,\\, \\left\\lvert \\beta_1 \\right\\rvert ,\\, \\cdots ,\\, \\left\\lvert \\beta_{N_1} \\right\\rvert \\right\\} +1\\]\n\uc774\ub77c\uace0 \ud558\uc790. \\(K\\)\uac00 \uc591\uc218\uc774\ubbc0\ub85c \uc790\uc5f0\uc218 \\(N_2\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(n \\ge N_2\\)\uc77c \ub54c\n\\[\\left\\lvert a_{n-N_1} \\right\\rvert < \\frac{\\epsilon}{2K \\left( N_1 +1 \\right)}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(N = \\max \\left\\{ N_1 ,\\, N_2 \\right\\}\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(n > N\\)\uc77c \ub54c<br \/>\n\\[\\begin{align}<br \/>\n\\left\\lvert \\gamma_n \\right\\rvert<br \/>\n&#038;\\le \\left\\lvert \\beta_0 a_n + \\beta_1 a_{n-1} + \\cdots + \\beta_{N_1} a_{n-N_1} \\right\\rvert + \\left\\lvert \\beta_{N_1 +1 } a_{n-N_1 -1} + \\cdots + \\beta_n a_0 \\right\\rvert \\\\[8pt]<br \/>\n&#038;\\le \\left( \\left\\lvert \\beta_0 \\right\\rvert \\left\\lvert a_n \\right\\rvert + \\cdots + \\left\\lvert \\beta_{N_1} \\right\\rvert \\left\\lvert a_{n-N_1} \\right\\rvert \\right) + \\left( \\left\\lvert \\beta_{N_1 +1} \\right\\rvert \\left\\lvert a_{n-N_1 &#8211; 1} \\right\\rvert + \\cdots + \\left\\lvert \\beta_n \\right\\rvert \\left\\lvert a_0 \\right\\rvert \\right) \\\\[6pt]<br \/>\n&#038;\\le K \\left( \\left\\lvert a_n \\right\\rvert + \\cdots + \\left\\lvert a_{n-N_1} \\right\\rvert \\right) + \\frac{\\epsilon}{2\\alpha +1} \\left( \\left\\lvert a_{n-N_1 -1} \\right\\rvert + \\cdots + \\left\\lvert a_0 \\right\\rvert \\right) \\\\[6pt]<br \/>\n&#038;\\le K\\left(N_1 +1 \\right) \\frac{\\epsilon}{2K \\left( N_1 +1 \\right)} + \\frac{\\epsilon}{2\\alpha + 1} \\alpha < \\frac{\\epsilon}{2} + \\frac{\\epsilon}{2} = \\epsilon\n\\end{align}\\]\n\uc774\ubbc0\ub85c \\(\\gamma_n \\to 0\\)\uc774\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"remark\">\n<p><span class=\"remark\">\ucc38\uace0.<\/span><br \/>\n\ub450 \ubb34\ud55c\uae09\uc218 \\(\\sum a_n\\)\uacfc \\(\\sum b_n\\)\uc774 \ubaa8\ub450 \uc870\uac74\uc218\ub834\ud560 \ub54c\uc5d0\ub294 \ucf54\uc2dc \uacf1 \\(\\sum c_n\\)\uc774 \ubc1c\uc0b0\ud560 \uc218\ub3c4 \uc788\ub2e4. \uc608\ucee8\ub300<br \/>\n\\[a_n = b_n = \\frac{(-1)^n}{\\sqrt{n+1}}\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uba74 \\(\\sum a_n\\)\uacfc \\(\\sum b_n\\)\uc740 \ubaa8\ub450 \uc870\uac74\uc218\ub834\ud55c\ub2e4. \uadf8\ub7f0\ub370<br \/>\n\\[\\begin{align}<br \/>\n\\left\\lvert c_n \\right\\rvert &#038;= \\left\\lvert \\sum_{k=0}^{n} \\frac{(-1)^k}{\\sqrt{k+1}} \\cdot \\frac{(-1)^{n-k}}{\\sqrt{n-k+1}} \\right\\rvert \\\\[2pt]<br \/>\n&#038;= \\left\\lvert (-1)^n \\sum_{k=0}^{n} \\frac{1}{\\sqrt{(k+1)(n-k+1)}} \\right\\rvert \\\\[2pt]<br \/>\n&#038;\\ge \\sum_{k=0}^{n}\\frac{1}{n+1} \\ge 1<br \/>\n\\end{align}\\]<br \/>\n\uc774\ubbc0\ub85c \\(\\left\\{ c_n \\right\\}\\)\uc740 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\sum c_n \\)\uc740 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n<p>\uadf8\ub807\ub2e4\uba74 \uc870\uac74\uc218\ub834\ud558\ub294 \ub450 \ubb34\ud55c\uae09\uc218\uc758 \ucf54\uc2dc \uacf1\uc740 \ud56d\uc0c1 \ubc1c\uc0b0\ud560\uae4c? \uadf8\ub807\uc9c0 \uc54a\ub2e4. \uc870\uac74\uc218\ub834\ud558\ub294 \ub450 \ubb34\ud55c\uae09\uc218\uc758 \ucf54\uc2dc \uacf1\uc774 \uc218\ub834\ud558\ub294 \uacbd\uc6b0\ub3c4 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7ec\ud55c \ubb34\ud55c\uae09\uc218\uc758 \uc608\ub294 \uc9c1\uc811 \ub9cc\ub4e4\uc5b4 \ubcf4\uae30 \ubc14\ub780\ub2e4. (\uc5b4\ub835\uc9c0 \uc54a\ub2e4.)<br \/>\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<p>\ubb34\ud55c\uae09\uc218\ub97c \ubb34\ud55c\uae09\uc218\ub85c \ub098\ub208 \uc2dd\uc740 \ud558\ub098\uc758 \ubb34\ud55c\uae09\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\uc744\uae4c? \uc870\uae08 \ub354 \ub2e8\uc21c\ud654\uc2dc\ucf1c \uc0dd\uac01\ud558\uc790\uba74, \uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\uc758 \uc5ed\uc218\ub97c \ubb34\ud55c\uae09\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\uc744\uae4c? \uc5ec\uae30\uc11c\ub294 \uac04\ub2e8\ud55c \uc544\uc774\ub514\uc5b4\ub9cc \uc18c\uac1c\ud558\uaca0\ub2e4.<\/p>\n<p>\\(\\sum a_n\\)\uc774 \uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d (8)\uc758 \ud45c\uae30\ubc95\uc5d0 \ub530\ub978 \ucf54\uc2dc \uacf1<br \/>\n\\[\\left( \\sum a_n \\right) \\left( \\sum b_n \\right) = \\sum c_n \\]<br \/>\n\uc758 \uac12\uc774 \\(1\\)\uc774 \ub418\ub3c4\ub85d \ud558\ub294 \ubb34\ud55c\uae09\uc218 \\(\\sum b_n\\)\uc744 \ucc3e\uc744 \uc218 \uc788\ub2e4\uba74 \uadf8\uac83\uc774 \uace7<br \/>\n\\[\\frac{1}{\\sum a_n} = \\sum b_n\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubb34\ud55c\uae09\uc218\uac00 \ub41c\ub2e4. \uc5ec\uae30\uc11c \\(\\sum a_n\\)\uacfc \\(\\sum b_n\\) \uc911 \ud558\ub098 \uc774\uc0c1\uc740 \uc808\ub300\uc218\ub834\ud574\uc57c \ud560 \uac83\uc774\ub2e4.<\/p>\n<p>\uadf8\ub7ec\ub098 \ub2e8\uc21c\ud788 \ubb34\ud55c\uae09\uc218\uc758 \uc5ed\uc218\ub97c \ubb34\ud55c\uae09\uc218\ub85c \ub098\ud0c0\ub0b4\ub294 \uac83\ubcf4\ub2e4\ub294 \uac70\ub4ed\uc81c\uacf1\uae09\uc218(power series)\uc758 \uc5ed\uc218\ub97c \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \ud45c\ud604\ud558\ub294 \uac83\uc774 \ub354 \uc4f8\ubaa8 \uc788\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc5ec\uae30\uc11c\ub294 \ubb34\ud55c\uae09\uc218\uc758 \uc5ed\uc218\ub97c \ubb34\ud55c\uae09\uc218\ub85c \ub098\ud0c0\ub0b4\ub294 \ubc29\ubc95\uc744 \ub354 \uc774\uc0c1 \uae4a\uc774 \uc0b4\ud3b4\ubcf4\uc9c0\ub294 \uc54a\uaca0\ub2e4.<\/p>\n<p><!--\n\n\n\n\n<div class=\"definition\">\n\n\n<p><span class=\"definition\">\uc815\uc758<\/span><\/p>\n\n\n\n\n<p> ... <\/p>\n\n<\/div>\n\n\n\n\n\n\n\n<div class=\"theorem\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac<\/span><\/p>\n\n\n\n\n<p> ... <\/p>\n\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85<\/p>\n\n\n\n\n<p>\n...\n<span class=\"qed\"><\/span><\/p>\n\n\n<\/div>\n\n\n\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 n.<\/span>\n...\n<span class=\"qee\"><\/span>\n<\/p>\n\n\n<\/div>\n\n\n\n\n\n\n<div class=\"remark\">\n\n\n<p><span class=\"remark\">\ucc38\uace0.<\/span>\n...\n<span class=\"qee\"><\/span>\n<\/p>\n\n\n<\/div>\n\n\n\n\n--><\/p>\n<p><!-- . --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\ubb34\ud55c\uae09\uc218\ub294 \uc624\ub798 \uc804\ubd80\ud130 \uc218\ud559\uc790\ub4e4\uc744 \ub2f9\ud639\uc2a4\ub7fd\uac8c \ub9cc\ub4e0 \uc8fc\uc81c \uc911 \ud558\ub098\uc600\ub2e4. \uc608\ucee8\ub300 \\(1 + \\frac{1}{2} + \\frac{1}{2^2} + \\frac{1}{2^3} + \\cdots\\) \ub294 \uc591\uc218\ub97c \ubb34\ud55c\ud788 \ub9ce\uc774 \ub354\ud568\uc5d0\ub3c4 \ubd88\uad6c\ud558\uace0 \uc218\ub834\ud558\uc9c0\ub9cc \\( 1 + \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\cdots\\) \uc740 \ubc1c\uc0b0\ud55c\ub2e4. \uc81c\ub17c\uc758 \uc5ed\uc124\ub3c4 \uace0\ub300\uc5d0 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ud559\uc790\ub4e4\uc744 \uc5bc\ub9c8\ub098 \uad34\ub86d\ud614\ub294\uc9c0\ub97c \ubcf4\uc5ec\uc8fc\ub294 \ubc29\uc99d\uc774\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubb34\ud55c\uae09\uc218\ub97c \uc815\uc758\ud558\uace0 \uc911\uc694\ud55c \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \ub0b4\uc6a9 \uc21c\uc11c \ubb34\ud55c\uae09\uc218\uc758 \ub73b \uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\uc758 \ub300\uc218\uc801 \uc5f0\uc0b0 \ubbf8\ub9ac \uc54c\uc544\uc57c&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[333,331,330,334,336,327,337,338,328,329,335,332],"class_list":["post-2018","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-cauchy-","tag-franz-mertens","tag-mertens-","tag-334","tag-336","tag-327","tag-337","tag-338","tag-328","tag-329","tag-335","tag-332"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2018","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=2018"}],"version-history":[{"count":52,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2018\/revisions"}],"predecessor-version":[{"id":9231,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2018\/revisions\/9231"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=2018"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=2018"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=2018"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}