{"id":2020,"date":"2019-06-21T13:15:00","date_gmt":"2019-06-21T04:15:00","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=2020"},"modified":"2023-11-22T22:40:36","modified_gmt":"2023-11-22T13:40:36","slug":"calculus-convergence-tests-of-series","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-convergence-tests-of-series\/","title":{"rendered":"\ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834 \ud310\uc815\ubc95"},"content":{"rendered":"<p>\ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud558\ub294\uc9c0 \ud310\ubcc4\ud558\ub294 \ubc29\ubc95\uc744 <span class=\"defined\">\uc218\ub834 \ud310\uc815\ubc95<\/span>(convergence test) \ub610\ub294 \uac04\ub2e8\ud788 <span class=\"defined\">\ud310\uc815\ubc95<\/span>(test)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubb34\ud55c\uae09\uc218\uc758 \ub2e4\uc591\ud55c \ud310\uc815\ubc95\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=\"#positiveterms\">\uc591\ud56d\uae09\uc218\uc758 \uc218\ub834 \ud310\uc815\ubc95<\/a><\/li>\n<li><a href=\"#absoluteconvergence\">\ubb34\ud55c\uae09\uc218\uc758 \uc808\ub300\uc218\ub834 \ud310\uc815\ubc95<\/a><\/li>\n<li><a href=\"#alternationtest\">\uad50\ub300\uae09\uc218 \ud310\uc815\ubc95<\/a><\/li>\n<li><a href=\"#cauchytest\">\ucf54\uc2dc\uc758 \uc751\uc9d1 \ud310\uc815\ubc95<\/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<li>\ubb34\ud55c\uae09\uc218 (<a href=\"\/blog\/articles\/calculus-infinite-series\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<\/ul>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"positiveterms\"><\/a><\/p>\n<h3>\uc591\ud56d\uae09\uc218\uc758 \uc218\ub834 \ud310\uc815\ubc95<\/h3>\n<p>\ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc778 \uc218\uc5f4\uc758 \ubb34\ud55c\uae09\uc218\ub97c <span class=\"defined\">\uc591\ud56d\uae09\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \uc591\ud56d\uae09\uc218\uc758 \ud310\uc815\ubc95\uc744 \uc774\uc6a9\ud558\uba74 \uc591\ud56d\uc774 \uc544\ub2cc \uae09\uc218\uc5d0 \ub300\ud574\uc11c\ub3c4 \uc808\ub300\uc218\ub834 \uc5ec\ubd80\ub97c \ud310\uc815\ud560 \uc218 \uc788\uae30 \ub54c\ubb38\uc5d0, \uc591\ud56d\uae09\uc218 \ud310\uc815\ubc95\uc740 \uc218\ub834 \ud310\uc815\ubc95\uc758 \uae30\ubcf8\uc774\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 1. (\uc720\uacc4 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc591\ud56d\uae09\uc218\uc77c \ub54c, \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uadf8 \ubd80\ubd84\ud569 \uc218\uc5f4 \\(\\sum_{k=1}^{n} a_k\\)\uac00 \uc720\uacc4\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uc218\ub834\ud558\ub294 \uc218\uc5f4\uc740 \uc720\uacc4\uc774\ubbc0\ub85c, \uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\uc758 \ubd80\ubd84\ud569 \uc218\uc5f4\uc740 \ub2f9\uc5f0\ud788 \uc720\uacc4\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c \uc5ed\uc744 \uc99d\uba85\ud558\uc790. \ubd80\ubd84\ud569 \uc218\uc5f4 \\(s_n = \\sum_{k=1}^{n} a_k\\)\uac00 \uc720\uacc4\ub77c\uace0 \uac00\uc815\ud558\uc790. \ubd80\ubd84\ud569 \uc218\uc5f4 \\(\\left\\{ s_n \\right\\}\\)\uc740 \uc99d\uac00\uc218\uc5f4\uc774\ubbc0\ub85c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ub2e4\uc74c \uc815\ub9ac\ub294 \uac10\uc18c\ud558\ub294 \uc591\ud56d\uc218\uc5f4\uc758 \ubb34\ud55c\uae09\uc218\uc640 \uc801\ubd84\uc758 \uad00\uacc4\ub97c \ubcf4\uc5ec\uc900\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 2. (\uc801\ubd84 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\\(N\\)\uc774 \uc0c1\uc218\uc774\uace0, \\(f\\)\uac00 \\([N,\\,\\infty )\\)\uc5d0\uc11c \uc815\uc758\ub41c \ub2e8\uc870\uac10\uc18c \ud568\uc218\uc774\uba70 \\(n \\ge N\\)\uc778 \ubaa8\ub4e0 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n = f(n)\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=N}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc774\uc0c1\uc801\ubd84<br \/>\n\\[\\int_{N}^{\\infty} f(x) dx\\tag{1}\\]<br \/>\n\uac00 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(k > N\\)\uc778 \uc790\uc5f0\uc218 \\(k\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[ a_{k+1} = \\int_{k}^{k+1} a_{k+1} \\,dx \\le \\int_{k}^{k+1} f(x) \\,dx \\le \\int_{k}^{k+1} a_k \\,dx = a_k\\tag{2}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[ \\sum_{k=N}^{n} a_{k+1} \\le \\int_{N}^{n+1} f(x)\\,dx \\le \\sum_{k=N}^{n} a_k \\tag{3}\\]<br \/>\n\uc774\ub2e4. \ub9cc\uc57d \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud558\uba74 (3)\uc758 \ub450 \ubc88\uc9f8 \ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud558\uc5ec \uc774\uc0c1\uc801\ubd84 (1)\ub3c4 \uc218\ub834\ud55c\ub2e4. \ub9cc\uc57d \uc774\uc0c1\uc801\ubd84 (1)\uc774 \uc218\ub834\ud558\uba74 (3)\uc758 \uccab \ubc88\uc9f8 \ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud558\uc5ec \\(\\sum_{n=1}^{\\infty} a_n\\)\uc740 \uc218\ub834\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span><br \/>\n\ubb34\ud55c\uae09\uc218<br \/>\n\\[\\sum_{n=1}^{\\infty} \\frac{1}{n^p}\\]<br \/>\n\uc774 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc744 \uad6c\ud574\ubcf4\uc790. \ub9cc\uc57d \\( p \\le 0\\)\uc774\uba74 \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\ub294 \ub2f9\uc5f0\ud788 \ubc1c\uc0b0\ud55c\ub2e4. \ub9cc\uc57d \\(p>0\\)\uc774\uba74 \uc774\uc0c1\uc801\ubd84<br \/>\n\\[\\int_{1}^{\\infty} \\frac{1}{x^p} dx\\]<br \/>\n\uac00 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(p > 1\\)\uc778 \uac83\uc774\ubbc0\ub85c, \uc801\ubd84 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \ubb34\ud55c\uae09\uc218 \\(\\sum \\frac{1}{n^p}\\)\uc774 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74 \ub610\ud55c \\(p > 1\\)\uc774 \ub41c\ub2e4.<\/p>\n<p>\uc774\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(\\sum \\frac{1}{n^p}\\)\uc758 \uc218\ub834\uc131\uc744 \ud310\ubcc4\ud558\ub294 \ud310\uc815\ubc95\uc744 <span class=\"defined\">p-\uae09\uc218 \ud310\uc815\ubc95<\/span>(p-series test)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<br \/>\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<p>\uc815\ub9ac 2\uc5d0\uc11c \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud558\uace0 \uadf8 \uac12\uc744 \\(S\\)\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc758 \ubd80\ubd84\ud569 \\(s_n = \\sum_{k=1}^{n} a_k\\)\uc5d0 \ub300\ud558\uc5ec \uc624\ucc28(remainder)\ub97c \\(R_n = S-s_n\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ubd80\ub4f1\uc2dd (2)\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc801\ubd84 \ud310\uc815\ubc95\uc5d0\uc11c \uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\uc758 \ub098\uba38\uc9c0\ud56d\uc758 \uc624\ucc28\uc758 \ud55c\uacc4 \uacf5\uc2dd<\/span><\/p>\n<p>\\[\\int_{n+1}^{\\infty} f(x) \\,dx \\le R_n \\le \\int_{n}^{\\infty} f(x) dx\\]<br \/>\n\ub2e8, \uc5ec\uae30\uc11c \\(n > N\\)\uc774\ub2e4.<\/p>\n<\/div>\n<p>\uc774\ubc88\uc5d0\ub294 \ub2e4\ub978 \ubb34\ud55c\uae09\uc218\uc640 \ube44\uad50\ud558\uc5ec \ud310\uc815\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 3. (\ube44\uad50 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\\(\\sum a_n\\)\uacfc \\(\\sum b_n\\)\uc774 \uc591\ud56d\uae09\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(n > N\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n \\le b_n\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74, \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\\(\\sum b_n\\)\uc774 \uc218\ub834\ud558\uba74 \\(\\sum a_n\\)\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/li>\n<li>\\(\\sum a_n\\)\uc774 \ubc1c\uc0b0\ud558\uba74 \\(\\sum b_n\\)\ub3c4 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>[1] \\(\\sum b_n\\)\uc774 \uc218\ub834\ud558\uba74 \\(\\sum a_n\\)\uc758 \ubd80\ubd84\ud569 \uc218\uc5f4\uc774 \uc720\uacc4\uc774\ubbc0\ub85c \uc815\ub9ac 1\uc5d0 \uc758\ud558\uc5ec \\(\\sum a_n\\)\uc740 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>[2] \\(\\sum a_n\\)\uc774 \ubc1c\uc0b0\ud558\uba74 \\(\\sum b_n\\)\uc758 \ubd80\ubd84\ud569 \uc218\uc5f4\uc740 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ubbc0\ub85c \uc815\ub9ac 1\uc5d0 \uc758\ud558\uc5ec \\(\\sum b_n\\)\uc740 \ubc1c\uc0b0\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 4. (\uadf9\ud55c \ube44\uad50 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(n > N\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n > 0,\\) \\(b_n > 0\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74, \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\\(a_n \/ b_n \\,\\to\\, c\\)\uc774\uace0 \\(0 < c < \\infty\\)\uc774\uba74, \\(\\sum a_n\\)\uacfc \\(\\sum b_n\\)\uc758 \uc218\ub834\uc131\uc740 \uc11c\ub85c \ub3d9\uce58\uc774\ub2e4.<\/li>\n<li>\\(a_n \/ b_n \\,\\to\\, 0\\)\uc774\uace0 \\(\\sum b_n\\)\uc774 \uc218\ub834\ud558\uba74 \\(\\sum a_n\\)\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/li>\n<li>\\(a_n \/ b_n \\,\\to\\, \\infty\\)\uc774\uace0 \\(\\sum b_n\\)\uc774 \ubc1c\uc0b0\ud558\uba74 \\(\\sum a_n\\)\ub3c4 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>[1] \\(\\epsilon = c\/2\\)\ub77c\uace0 \ub450\uba74, \\(N\\)\ubcf4\ub2e4 \ub354 \ud070 \uc790\uc5f0\uc218 \\(N_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N_1\\)\uc77c \ub54c\ub9c8\ub2e4<br \/>\n\\[ c &#8211; \\epsilon < \\frac{a_n}{b_n} < c+ \\epsilon\\]\n\uc989\n\\[\\frac{c}{2} b_n < a_n < \\frac{3c}{2} b_n\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc774 \ubd80\ub4f1\uc2dd\uc744 \uc774\uc6a9\ud558\uba74 \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(\\sum a_n\\)\uc758 \uc218\ub834\uc131\uacfc \\(\\sum b_n\\)\uc758 \uc218\ub834\uc131\uc774 \ub3d9\uce58\ub77c\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<p>[2] \\(\\epsilon = 1\\)\uc774\ub77c\uace0 \ub450\uba74, \\(N\\)\ubcf4\ub2e4 \ub354 \ud070 \uc790\uc5f0\uc218 \\(N_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N_1\\)\uc77c \ub54c\ub9c8\ub2e4<br \/>\n\\[0-\\epsilon < \\frac{a_n}{b_n} < 0+\\epsilon\\]\n\uc989\n\\[ -b_n < a_n < b_n\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c, \ub9cc\uc57d \\(\\sum b_n\\)\uc774 \uc218\ub834\ud558\uba74 \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(\\sum a_n\\)\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>[3] \\(a_n \/ b_n \\,\\to\\,\\infty\\)\uc774\ubbc0\ub85c, \\(N\\)\ubcf4\ub2e4 \ub354 \ud070 \uc790\uc5f0\uc218 \\(N_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N_1\\)\uc77c \ub54c\ub9c8\ub2e4<br \/>\n\\[\\frac{a_n}{b_n} > 1\\]<br \/>\n\uc989<br \/>\n\\[a_n > b_n\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c, \ub9cc\uc57d \\(\\sum b_n\\)\uc774 \ubc1c\uc0b0\ud558\uba74 \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(\\sum a_n\\)\ub3c4 \ubc1c\uc0b0\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"absoluteconvergence\"><\/a><\/p>\n<h3>\ubb34\ud55c\uae09\uc218\uc758 \uc808\ub300\uc218\ub834 \ud310\uc815\ubc95<\/h3>\n<p>\uc591\ud56d\uae09\uc218\uc758 \ud310\uc815\ubc95\uc744 \uc774\uc6a9\ud558\uba74 \uc8fc\uc5b4\uc9c4 \uae09\uc218\uac00 \uc808\ub300\uc218\ub834\ud558\ub294\uc9c0 \uc5ec\ubd80\ub97c \ud310\uc815\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\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>\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>\ud55c\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 5.<\/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<a href=\"\/blog\/articles\/calculus-infinite-series\">\ubb34\ud55c\uae09\uc218<\/a> \ud3ec\uc2a4\ud2b8\uc758 \uc815\ub9ac 2\uc5d0\uc11c \uc99d\uba85\ud558\uc600\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc774\uc81c \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\uac00 \uc808\ub300\uc218\ub834\ud558\ub294\uc9c0\ub97c \ud310\uc815\ud558\ub294 \uac00\uc7a5 \ub300\ud45c\uc801\uc778 \ub450 \uac1c\uc758 \ud310\uc815\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc790. \uc774 \ub450 \ud310\uc815\ubc95\uc740 \ub4a4\uc5d0\uc11c \uac70\ub4ed\uc81c\uacf1\uae09\uc218\uc758 \uc218\ub834\ubc18\uacbd\uc744 \uad6c\ud560 \ub54c \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 6. (\ube44 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\\(\\sum a_n\\)\uc774 \ubb34\ud55c\uae09\uc218\uc774\uace0, \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n > N\\)\uc77c \ub54c \\(a_n \\ne 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0<br \/>\n\\[\\lim_{n\\to\\infty} \\left\\lvert \\frac{a_{n+1}}{a_n} \\right\\rvert = \\rho\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\ub9cc\uc57d \\(\\rho < 1\\)\uc774\uba74 \\(\\sum a_n\\)\uc740 \uc808\ub300\uc218\ub834\ud55c\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(\\rho > 1\\)\uc774\uba74 \\(\\sum a_n\\)\uc740 \ubc1c\uc0b0\ud55c\ub2e4. (\uc5ec\uae30\uc11c \\(\\rho\\)\ub294 \ubb34\ud55c\ub300\uc77c \uc218\ub3c4 \uc788\ub2e4.)<\/li>\n<li>\ub9cc\uc57d \\(\\rho = 1\\)\uc774\uba74 \ube44 \ud310\uc815\ubc95\uc73c\ub85c \ud310\uc815\ud560 \uc218 \uc5c6\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\ud45c\uae30\ub97c \uac04\ub2e8\ud558\uac8c \ud558\uae30 \uc704\ud574 \\(b_n = \\left\\lvert a_n \\right\\rvert\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<p>[1] \\(\\epsilon = (1-\\rho) \/ 2\\)\ub77c\uace0 \ud558\uc790. \\(b_{n_1} \/ b_n \\to \\rho\\)\uc774\ubbc0\ub85c \\(N\\)\ubcf4\ub2e4 \ud070 \uc790\uc5f0\uc218 \\(N_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(k \\ge N_1\\)\uc77c \ub54c<br \/>\n\\[\\frac{b_{k+1}}{b_k} < \\rho + \\epsilon\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(n > N_1\\)\uc77c \ub54c<br \/>\n\\[b_n = \\frac{b_n}{b_{n-1}} \\frac{b_{n-1}}{b_{n-2}} \\cdots \\frac{b_{N_1 +1}}{b_{N_1}} b_{N_1}<br \/>\n < (\\rho + \\epsilon)^{n-N_1} b_{N_1}\\]\n\uc774\ubbc0\ub85c\n\\[\\sum_{k=N_1 +1}^{n} b_k < \\sum_{k=N_1 +1}^{n} (\\rho + \\epsilon)^{k-N_1} b_{N_1}\\]\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(n\\to\\infty\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74, \ubd80\ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc740 \uacf5\ube44\uc758 \uc808\ub313\uac12\uc774 \\(1\\)\ubcf4\ub2e4 \uc791\uc740 \ub4f1\ube44\uc218\uc5f4\uc758 \ud569\uc774\ubbc0\ub85c \uc218\ub834\ud558\uba70, \uc591\ud56d\uae09\uc218\uc758 \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \uc88c\ubcc0\ub3c4 \uc218\ub834\ud55c\ub2e4. \uc989 \\(\\sum b_n\\)\uc774 \uc218\ub834\ud558\ubbc0\ub85c \\(\\sum a_n\\)\uc740 \uc808\ub300\uc218\ub834\ud55c\ub2e4.\n<\/p>\n<p>[2] \\(\\left\\lvert a_n \\right\\rvert \\to \\infty\\)\uc774\ubbc0\ub85c \\(\\sum a_n\\)\uc740 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n<p>[3] \\(\\sum (1\/n)\\)\uacfc \\(\\sum \\left( 1\/n^2 \\right)\\)\uc740 \ubaa8\ub450 \ube44 \ud310\uc815\ubc95\uc744 \uc801\uc6a9\ud588\uc744 \ub54c \\(\\rho = 1\\)\uc778 \ubb34\ud55c\uae09\uc218\uc774\uc9c0\ub9cc \uc55e\uc758 \uac83\uc740 \ubc1c\uc0b0\ud558\uace0 \ub4a4\uc758 \uac83\uc740 \uc218\ub834\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 7. (\uc81c\uacf1\uadfc \ud310\uc815\ubc95)<\/span><\/p>\n<p>\\(\\sum a_n\\)\uc774 \ubb34\ud55c\uae09\uc218\uc774\uace0<br \/>\n\\[\\lim_{n\\to\\infty} \\sqrt[n]{\\left\\lvert a_n \\right\\rvert} = \\rho\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\ub9cc\uc57d \\(\\rho < 1\\)\uc774\uba74 \\(\\sum a_n\\)\uc740 \uc808\ub300\uc218\ub834\ud55c\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(\\rho > 1\\)\uc774\uba74 \\(\\sum a_n\\)\uc740 \ubc1c\uc0b0\ud55c\ub2e4. (\uc5ec\uae30\uc11c \\(\\rho\\)\ub294 \ubb34\ud55c\ub300\uc77c \uc218\ub3c4 \uc788\ub2e4.)<\/li>\n<li>\ub9cc\uc57d \\(\\rho = 1\\)\uc774\uba74 \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc73c\ub85c \ud310\uc815\ud560 \uc218 \uc5c6\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\ud45c\uae30\ub97c \uac04\ub2e8\ud558\uac8c \ud558\uae30 \uc704\ud574 \\(b_n = \\left\\lvert a_n \\right\\rvert\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<p>[1] \\(\\epsilon = (1-\\rho) \/2\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(k \\ge N\\)\uc77c \ub54c<br \/>\n\\[\\sqrt[k]{b_k} < \\rho + \\epsilon\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc591\ubcc0\uc744 \\(k\\)\uc81c\uacf1\ud558\uba74\n\\[b_k < (\\rho + \\epsilon)^k\\]\n\uc744 \uc5bb\ub294\ub2e4. \\(n > N\\)\uc77c \ub54c<br \/>\n\\[\\sum_{k=N}^{n} b_k < \\sum_{k=N}^{n} (\\rho + \\epsilon)^k\\]\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(n\\to\\infty\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74, \ubd80\ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc740 \uacf5\ube44\uc758 \uc808\ub313\uac12\uc774 \\(1\\)\ubcf4\ub2e4 \uc791\uc740 \ub4f1\ube44\uc218\uc5f4\uc758 \ud569\uc774\ubbc0\ub85c \uc218\ub834\ud558\uba70, \uc591\ud56d\uae09\uc218\uc758 \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \uc88c\ubcc0\ub3c4 \uc218\ub834\ud55c\ub2e4. \uc989 \\(\\sum b_n\\)\uc774 \uc218\ub834\ud558\ubbc0\ub85c \\(\\sum a_n\\)\uc740 \uc808\ub300\uc218\ub834\ud55c\ub2e4.\n<\/p>\n<p>[2] \\(\\left\\lvert a_n \\right\\rvert \\to \\infty\\)\uc774\ubbc0\ub85c \\(\\sum a_n\\)\uc740 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n<p>[3] \\(\\sum (1\/n)\\)\uacfc \\(\\sum \\left( 1\/n^2 \\right)\\)\uc740 \ubaa8\ub450 \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc744 \uc801\uc6a9\ud588\uc744 \ub54c \\(\\rho = 1\\)\uc778 \ubb34\ud55c\uae09\uc218\uc774\uc9c0\ub9cc \uc55e\uc758 \uac83\uc740 \ubc1c\uc0b0\ud558\uace0 \ub4a4\uc758 \uac83\uc740 \uc218\ub834\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"alternationtest\"><\/a><\/p>\n<h3>\uad50\ub300\uae09\uc218 \ud310\uc815\ubc95<\/h3>\n<p>\uc591\uc218\uc778 \ud56d\uacfc \uc74c\uc218\uc778 \ud56d\uc774 \ubc88\uac08\uc544 \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 \\(\\sum_{n=1}^{\\infty} a_n \\)\uc740 \\(u_n = \\left\\lvert a_n \\right\\rvert\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n = \\sum_{n=1}^{\\infty} (-1)^n \\,u_n\\tag{4}\\]<br \/>\n\ub610\ub294<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n = \\sum_{n=1}^{\\infty} (-1)^{n+1} \\,u_n\\tag{5}\\]<br \/>\n\uc758 \uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. (4)\uc640 (5)\uc758 \uc218\ub834\uc131\uc740 \ub3d9\uc77c\ud558\ubbc0\ub85c, \uad50\ub300\uae09\uc218\uc758 \uc218\ub834\uc131\uc744 \ub17c\ud560 \ub54c\uc5d0\ub294 (5)\uc640 \uac19\uc740 \ud615\ud0dc\ub9cc \uc0b4\ud3b4\ubcf4\uc544\ub3c4 \ucda9\ubd84\ud558\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 8. (\uad50\ub300\uae09\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\\(\\left\\{ u_n \\right\\}\\)\uc774 \ub2e8\uc870\uac10\uc18c\ud558\ub294 \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \uad50\ub300\uae09\uc218<br \/>\n\\[\\sum_{n=1}^{\\infty} (-1)^{n+1} \\,u_n\\tag{6}\\]<br \/>\n\uc774 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740<br \/>\n\\[\\lim_{n\\to\\infty} u_n =0\\tag{7}\\]<br \/>\n\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\sum (-1)^{n+1} \\,u_n\\)\uc774 \uc218\ub834\ud558\uba74 \ub2f9\uc5f0\ud788 \\(u_n \\,\\to\\,0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc5ed\ub9cc \uc99d\uba85\ud558\uba74 \ub41c\ub2e4.<\/p>\n<p>\\(u_n \\,\\to\\, 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0<br \/>\n\\[s_n = \\sum_{k=1}^{n} (-1)^{k+1}\\,u_k\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uc774\uc81c \uc218\uc5f4 \\(\\left\\{ s_n \\right\\}\\)\uc774 \uc218\ub834\ud568\uc744 \ubcf4\uc5ec\uc57c \ud55c\ub2e4. \uc9dd\uc218\uc9f8 \ud56d\uacfc \ud640\uc218\uc9f8 \ud56d\uc774 \uac01\uac01 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uc790.<br \/>\n\\[\\begin{align}<br \/>\ns_{2m} &#038;= ( u_1 &#8211; u_2 ) + (u_3 &#8211; u_4) + \\cdots + (u_{2m-1} -u_{2m}) \\tag{8}\\\\[8pt]<br \/>\n&#038;= u_1 &#8211; (u_2 &#8211; u_3) &#8211; (u_4 &#8211; u_5 ) &#8211; \\cdots &#8211; (u_{2m-2} &#8211; u_{2m-1}) &#8211; u_{2m} \\tag{9}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. (8)\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{ s_{2m} \\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\uc218\uc5f4\uc774\uace0, (9)\uc5d0 \uc758\ud558\uc5ec \\(s_{2m} \\le u_1\\)\uc774\ubbc0\ub85c \\(\\left\\{ s_{2m} \\right\\}\\)\uc740 \uc704\ub85c \uc720\uacc4\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{ s_{2m} \\right\\}\\)\uc740 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\ube44\uc2b7\ud55c \ubc29\ubc95\uc73c\ub85c \\(\\left\\{ s_{2m+1} \\right\\}\\)\uc740 \ub2e8\uc870\uac10\uc18c\ud558\uace0 \uc544\ub798\ub85c \uc720\uacc4\uc774\ubbc0\ub85c \uc218\ub834\ud568\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4.<\/p>\n<p>\\(\\left\\{ s_{2m} \\right\\}\\)\uc758 \uadf9\ud55c\uac12\uc744 \\(L\\)\uc774\ub77c\uace0 \ud558\uace0, \\(\\left\\{ s_{2m+1} \\right\\}\\)\uc758 \uadf9\ud55c\uac12\uc744 \\(L_2\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[L_2 &#8211; L_1 = \\lim_{m\\to\\infty} (s_{2m+1} &#8211; s_{2m} ) = \\lim_{m\\to\\infty} u_{2m+1} = 0\\tag{10}\\]<br \/>\n\uc774\ubbc0\ub85c \\(L_1 = L_2\\)\uc774\ub2e4. \uc989 \\(\\left\\{ s_{2m} \\right\\}\\)\uacfc \\(\\left\\{ s_{2m+1} \\right\\}\\)\uc774 \uac19\uc740 \uac12\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c \\(\\left\\{ s_{n} \\right\\}\\)\ub3c4 \uc218\ub834\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\\(\\sum_{n=1}^{\\infty} (-1)^{n+1} \\,u_n = S\\)\ub77c\uace0 \ud558\uace0, \uadf8 \ubd80\ubd84\ud569\uc744 \\(s_n\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc815\ub9ac 8\uc758 \uc99d\uba85 \uacfc\uc815\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{ s_{2m} \\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\uc218\uc5f4\uc774\uace0 \\(\\left\\{ s_{2m+1} \\right\\}\\)\uc740 \ub2e8\uc870\uac10\uc18c\uc218\uc5f4\uc774\ubbc0\ub85c<br \/>\n\\[s_{2m} \\le S \\le s_{2m+1}\\]<br \/>\n\uc774\uba70, (10)\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[s_{2m+1} &#8211; s_{2m} = u_{2m+1}\\]<br \/>\n\uc774\ubbc0\ub85c \ub450 \uc2dd\uc744 \uacb0\ud569\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"box\">\n<p><span class=\"theorem\">\uad50\ub300\uae09\uc218\uc758 \ubd80\ubd84\ud569\uc758 \uc624\ucc28\uc758 \ud55c\uacc4 \uacf5\uc2dd<\/span><\/p>\n<p>\\[\\left\\lvert s_n &#8211; S \\right\\rvert \\le u_{n+1}\\]\n<\/p><\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"cauchytest\"><\/a><\/p>\n<h3>\ucf54\uc2dc\uc758 \uc751\uc9d1 \ud310\uc815\ubc95<\/h3>\n<p>\ub2e4\uc74c \ud310\uc815\ubc95\uc740 \ub85c\uadf8\uac00 \ud3ec\ud568\ub41c \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834\uc131\uc744 \ud310\uc815\ud560 \ub54c \uc720\uc6a9\ud558\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 9. (\ucf54\uc2dc\uc758 \uc751\uc9d1 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\\(\\left\\{ a_n \\right\\}\\)\uc774 \ub2e8\uc870\uac10\uc18c\ud558\uace0 \\(a_n \\ge 0\\)\uc778 \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub450 \ubb34\ud55c\uae09\uc218<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n ,\\quad \\sum_{n=1}^{\\infty} 2^n \\,a_{2^n}\\]<br \/>\n\uc758 \uc218\ub834\uc131\uc740 \uc11c\ub85c \ub3d9\uce58\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\sum a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \\(\\left\\{ a_n \\right\\}\\)\uc774 \ub2e8\uc870\uac10\uc18c\ud558\ubbc0\ub85c<br \/>\n\\[\\begin{align}<br \/>\n2a_4 &#038;= a_4 + a_4 \\le a_3 + a_4 ,\\\\[8pt]<br \/>\n4a_8 &#038;= a_8 + a_8 + a_8 + a_8 \\le a_5 + a_6 + a_7 + a_8 ,\\\\[8pt]<br \/>\n8a_{16} &#038;\\le a_9 + a_{10} + a_{11} + \\cdots + a_{16} ,\\\\[8pt]<br \/>\n16a_{32} &#038;\\le a_{17} + a_{18} + a_{19} + \\cdots + a_{32} ,\\\\[8pt]<br \/>\n&#038; \\,\\,\\vdots<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(k\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[2^k \\,a_{2^{k+1}} \\le a_{2^k +1} + a_{2^k +2} + a_{2^k +3} + \\cdots + a_{2^{k+1}}\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(k=1\\)\uc77c \ub54c\ubd80\ud130 \\(k=n\\)\uc77c \ub54c\uae4c\uc9c0 \ubd80\ub4f1\uc2dd\uc744 \ubcc0\ub9c8\ub2e4 \ub354\ud558\uba74<br \/>\n\\[\\frac{1}{2}\\sum_{k=1}^{n} 2^{k+1} \\,a_{2^{k+1}} \\le \\sum_{k=3}^{2^{n+1}} a_k\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4. \\(n\\to\\infty\\)\uc77c \ub54c \uc6b0\ubcc0\uc774 \uc218\ub834\ud558\ubbc0\ub85c \uc88c\ubcc0\uc740 \uc704\ub85c \uc720\uacc4\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc720\uacc4 \ud310\uc815\ubc95(\ub610\ub294 \ub2e8\uc870\uc218\ub834 \uc815\ub9ac)\uc5d0 \uc758\ud558\uc5ec \\(n\\to\\infty\\)\uc77c \ub54c \uc88c\ubcc0\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\uc5ed\uc744 \uc99d\uba85\ud558\uc790. \\(\\sum 2^{n}\\,a_{2^n}\\)\uc774 \uc218\ub834\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \\(\\left\\{ a_n \\right\\}\\)\uc774 \ub2e8\uc870\uac10\uc18c\ud558\ubbc0\ub85c<br \/>\n\\[\\begin{align}<br \/>\n1a_1 &#038;\\ge a_1 ,\\\\[8pt]<br \/>\n2a_2 &#038;= a_2 + a_2 \\ge a_2 +a_3 ,\\\\[8pt]<br \/>\n4a_4 &#038;= a_4 + a_4 + a_4 +a_4 \\ge a_4 + a_5 + a_6 + a_7 ,\\\\[8pt]<br \/>\n8a_8 &#038;\\ge a_8 + a_9 + a_{10} + \\cdots + a_{15}, \\\\[8pt]<br \/>\n&#038; \\,\\,\\vdots<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(k\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[2^k\\,a_{2^k} \\ge a_{2^k} + a_{2^k +1} + a_{2^k +2} + \\cdots + a_{2^{k+1}-1}\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(k=0\\)\uc77c \ub54c\ubd80\ud130 \\(k=n\\)\uc77c \ub54c\uae4c\uc9c0 \ubd80\ub4f1\uc2dd\uc744 \ubcc0\ub9c8\ub2e4 \ub354\ud558\uba74<br \/>\n\\[\\sum_{k=0}^n 2^k \\,a_{2^k} \\ge \\sum_{k=2}^{2^{n+1}-1} a_k\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4. \\(n\\to\\infty\\)\uc77c \ub54c \uc88c\ubcc0\uc774 \uc218\ub834\ud558\ubbc0\ub85c \uc6b0\ubcc0\uc740 \uc704\ub85c \uc720\uacc4\uc774\ub2e4. \uc989 \\(\\sum a_n\\)\uc758 \ubd80\ubd84\ud569 \\(s_n\\)\uc758 \ubd80\ubd84\uc218\uc5f4\uc774 \uc704\ub85c \uc720\uacc4\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(\\left\\{ s_n \\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud558\ub294 \uc218\uc5f4\uc774\ubbc0\ub85c \ud55c \ubd80\ubd84\uc218\uc5f4\uc774 \uc720\uacc4\uc774\uae30\ub9cc \ud558\uba74 \\(\\left\\{ s_n \\right\\}\\)\ub3c4 \uc720\uacc4\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc720\uacc4 \ud310\uc815\ubc95(\ub610\ub294 \ub2e8\uc870\uc218\ub834 \uc815\ub9ac)\uc5d0 \uc758\ud558\uc5ec \\(\\sum a_n\\)\uc740 \uc218\ub834\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ucf54\uc2dc\uc758 \uc751\uc9d1 \ud310\uc815\ubc95\uc740 \u2018\\(2^n\\)-\ud310\uc815\ubc95\u2019\uc774\ub77c\uace0\ub3c4 \ubd88\ub9b0\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\uac00 \uc218\ub834\ud558\ub294\uc9c0 \ud310\ubcc4\ud558\ub294 \ubc29\ubc95\uc744 \uc218\ub834 \ud310\uc815\ubc95(convergence test) \ub610\ub294 \uac04\ub2e8\ud788 \ud310\uc815\ubc95(test)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubb34\ud55c\uae09\uc218\uc758 \ub2e4\uc591\ud55c \ud310\uc815\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \ub0b4\uc6a9 \uc21c\uc11c \uc591\ud56d\uae09\uc218\uc758 \uc218\ub834 \ud310\uc815\ubc95 \ubb34\ud55c\uae09\uc218\uc758 \uc808\ub300\uc218\ub834 \ud310\uc815\ubc95 \uad50\ub300\uae09\uc218 \ud310\uc815\ubc95 \ucf54\uc2dc\uc758 \uc751\uc9d1 \ud310\uc815\ubc95 \ubbf8\ub9ac \uc54c\uc544\uc57c \ud560 \ub0b4\uc6a9 \uc218\uc5f4\uc758 \uadf9\ud55c (\uad00\ub828 \uae00) \ubb34\ud55c\uae09\uc218 (\uad00\ub828 \uae00) \uc591\ud56d\uae09\uc218\uc758 \uc218\ub834 \ud310\uc815\ubc95 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc778 \uc218\uc5f4\uc758 \ubb34\ud55c\uae09\uc218\ub97c \uc591\ud56d\uae09\uc218\ub77c\uace0 \ubd80\ub978\ub2e4. \uc591\ud56d\uae09\uc218\uc758 \ud310\uc815\ubc95\uc744 \uc774\uc6a9\ud558\uba74 \uc591\ud56d\uc774 \uc544\ub2cc \uae09\uc218\uc5d0 \ub300\ud574\uc11c\ub3c4 \uc808\ub300\uc218\ub834 \uc5ec\ubd80\ub97c \ud310\uc815\ud560 \uc218 \uc788\uae30&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":[348,351,352,346,349,350,347,344,345,343,341,327,340,342,328,329,339],"class_list":["post-2020","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-absolutely-convergence","tag-alternating-series","tag-alternating-series-test","tag-comparison-test","tag-conditionally-convergence","tag-convergence-test","tag-limit-comparison-test","tag-ratio-test","tag-root-test","tag-343","tag-341","tag-327","tag-340","tag-342","tag-328","tag-329","tag-339"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2020","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=2020"}],"version-history":[{"count":46,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2020\/revisions"}],"predecessor-version":[{"id":8937,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2020\/revisions\/8937"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=2020"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=2020"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=2020"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}