{"id":6686,"date":"2021-07-20T23:55:28","date_gmt":"2021-07-20T14:55:28","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6686"},"modified":"2026-09-27T17:28:38","modified_gmt":"2026-09-27T08:28:38","slug":"absolute-convergence","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/absolute-convergence\/","title":{"rendered":"\uc808\ub300\uc218\ub834\uacfc \uc870\uac74\uc218\ub834"},"content":{"rendered":"<div class=\"box itc_intro\">\n<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 2\uc7a5 3\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; <span class=\"itc_viewcontents\">(<a href=\"..\/\">\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n<p>\\(\\left\\{a_n\\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \ubb34\ud55c\uae09\uc218\uc758 \uc808\ub300\uc218\ub834\uacfc \uc870\uac74\uc218\ub834\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\ubb34\ud55c\uae09\uc218<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}|a_n|<br \/>\n\\]<br \/>\n\uc774 \uc218\ub834\ud558\uba74 \ubb34\ud55c\uae09\uc218<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_n<br \/>\n\\]<br \/>\n\uc774 <span class=\"defined\">\uc808\ub300\uc218\ub834<\/span>\ud55c\ub2e4(converge absolutely)\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<li>\ubb34\ud55c\uae09\uc218<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_n<br \/>\n\\]<br \/>\n\uc774 \uc218\ub834\ud558\uc9c0\ub9cc \uc808\ub300\uc218\ub834\ud558\uc9c0 \uc54a\uc73c\uba74 \uc774 \uae09\uc218\uac00 <span class=\"defined\">\uc870\uac74\uc218\ub834<\/span>\ud55c\ub2e4(converge conditionally)\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<\/ul>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc808\ub300\uc218\ub834\uacfc \uc870\uac74\uc218\ub834\uc758 \uad00\uacc4<\/h2>\n<p>\\(\\left\\{a_n\\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \\(a_n\\)\uc758 <span class=\"defined\">\uc591\uc758 \ubd80\ubd84<\/span>(positive part) \\(a_n^+\\)\uacfc <span class=\"defined\">\uc74c\uc758 \ubd80\ubd84<\/span>(negative part) \\(a_n^-\\)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\na_n^+<br \/>\n&#038;=<br \/>\n\\begin{cases}<br \/>\na_n &#038; \\quad\\text{if } a_n\\ge0,\\\\<br \/>\n0 &#038; \\quad\\text{if } a_n < 0,\n\\end{cases}\n\\\\[6pt]\na_n^-\n&#038;=\n\\begin{cases}\n-a_n &#038; \\quad\\text{if } a_n\\le0,\\\\\n0 &#038; \\quad\\text{if } a_n>0.<br \/>\n\\end{cases}<br \/>\n\\end{align}<br \/>\n\\]<br \/>\n\uadf8\ub7ec\uba74 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\na_n=a_n^+-a_n^-,<br \/>\n\\qquad<br \/>\n|a_n|=a_n^++a_n^-<br \/>\n\\]<br \/>\n\uc774\uace0<br \/>\n\\[<br \/>\n0\\le a_n^+\\le|a_n|,<br \/>\n\\qquad<br \/>\n0\\le a_n^-\\le|a_n|<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/p>\n<div class=\"lemma margintop2\">\n<p><span class=\"theorem\">\ubcf4\uc870\uc815\ub9ac 2.3.1.<\/span><br \/>\n\ubb34\ud55c\uae09\uc218<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_n<br \/>\n\\]<br \/>\n\uc774 \uc808\ub300\uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_n^+<br \/>\n\\quad\\text{\uc640}\\quad<br \/>\n\\sum_{n=1}^{\\infty}a_n^-<br \/>\n\\]<br \/>\n\uac00 \ubaa8\ub450 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\uba3c\uc800<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}|a_n|<br \/>\n\\]<br \/>\n\uc774 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n0\\le a_n^+\\le|a_n|,<br \/>\n\\qquad<br \/>\n0\\le a_n^-\\le|a_n|<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_n^+,<br \/>\n\\qquad<br \/>\n\\sum_{n=1}^{\\infty}a_n^-<br \/>\n\\]<br \/>\n\uac00 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\uac70\uafb8\ub85c \uc774 \ub450 \uae09\uc218\uac00 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[<br \/>\n|a_n|=a_n^++a_n^-<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \ubb34\ud55c\uae09\uc218\uc758 \uc120\ud615\uc131\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}|a_n|<br \/>\n=<br \/>\n\\sum_{n=1}^{\\infty}a_n^+<br \/>\n+<br \/>\n\\sum_{n=1}^{\\infty}a_n^-<br \/>\n\\]<br \/>\n\ub3c4 \uc218\ub834\ud55c\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_n<br \/>\n\\]<br \/>\n\uc740 \uc808\ub300\uc218\ub834\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 2.3.2. (\uc808\ub300\uc218\ub834 \ud310\uc815)<\/span><\/p>\n<p>\ubb34\ud55c\uae09\uc218<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_n<br \/>\n\\]<br \/>\n\uc774 \uc808\ub300\uc218\ub834\ud558\uba74 \uc774 \uae09\uc218\ub294 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\ubcf4\uc870\uc815\ub9ac 2.3.1\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_n^+<br \/>\n\\quad\\text{\uc640}\\quad<br \/>\n\\sum_{n=1}^{\\infty}a_n^-<br \/>\n\\]<br \/>\n\uac00 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7f0\ub370<br \/>\n\\[<br \/>\na_n=a_n^+-a_n^-<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \ubb34\ud55c\uae09\uc218\uc758 \uc120\ud615\uc131\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_n<br \/>\n=<br \/>\n\\sum_{n=1}^{\\infty}a_n^+<br \/>\n&#8211;<br \/>\n\\sum_{n=1}^{\\infty}a_n^-<br \/>\n\\]<br \/>\n\uc774 \uc218\ub834\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><span class=\"remark\">\ucc38\uace0.<\/span><br \/>\n\uc815\ub9ac 2.3.2\uc758 \uc5ed\uc740 \uc77c\ubc18\uc801\uc73c\ub85c \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4. \ub2e4\uc74c \uc808\uc5d0\uc11c<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}\\frac{(-1)^{n-1}}{n}<br \/>\n\\]<br \/>\n\uc774 \uc218\ub834\ud568\uc744 \ubcf4\uc77c \uac83\uc774\ub2e4. \uadf8\ub7ec\ub098<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}<br \/>\n\\left|<br \/>\n\\frac{(-1)^{n-1}}{n}<br \/>\n\\right|<br \/>\n=<br \/>\n\\sum_{n=1}^{\\infty}\\frac1n<br \/>\n\\]<br \/>\n\uc740 \ubc1c\uc0b0\ud558\ubbc0\ub85c \uc774 \uae09\uc218\ub294 \uc870\uac74\uc218\ub834\ud55c\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2.3.1.<\/span><br \/>\n\ub2e4\uc74c \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud558\ub3c4\ub85d \ud558\ub294 \uc2e4\uc218 \\(x\\)\uc758 \ubc94\uc704\ub97c \uad6c\ud558\uc2dc\uc624.<br \/>\n\\[<br \/>\n1+2x+x^2+2x^3+x^4+2x^5+\\cdots.<br \/>\n\\]\n<\/p>\n<p><span class=\"proof\">\ud480\uc774.<\/span><br \/>\n\\[<br \/>\n|x| < 1\n\\]\n\uc774\ub77c\uace0 \ud558\uc790. \uc8fc\uc5b4\uc9c4 \uae09\uc218\uc758 \uac01 \ud56d\uc758 \uc808\ub313\uac12\uc744 \ucde8\ud55c \uae09\uc218\ub294\n\\[\n1+2|x|+|x|^2+2|x|^3+|x|^4+2|x|^5+\\cdots\n\\]\n\uc774\ub2e4. \uc774\ub97c \ub450 \ub4f1\ube44\uae09\uc218\ub85c \ub098\ub204\uba74\n\\[\n\\sum_{k=0}^{\\infty}|x|^{2k}\n+\n2|x|\n\\sum_{k=0}^{\\infty}|x|^{2k}\n\\]\n\uac00 \ub41c\ub2e4. \\(|x|^2 < 1\\)\uc774\ubbc0\ub85c \ub450 \ub4f1\ube44\uae09\uc218\uac00 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4. \ub530\ub77c\uc11c \uc6d0\ub798 \uae09\uc218\ub294 \uc808\ub300\uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\ubc18\ub300\ub85c<br \/>\n\\[<br \/>\n|x|\\ge1<br \/>\n\\]<br \/>\n\uc774\uba74 \uc6d0\ub798 \uae09\uc218\uc758 \ud56d\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c \uc77c\ubc18\ud56d \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \uae09\uc218\ub294 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n<p>\ub530\ub77c\uc11c \uc8fc\uc5b4\uc9c4 \uae09\uc218\uac00 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740<br \/>\n\\[<br \/>\n-1 < x < 1\n\\]\n\uc774\ub2e4.<\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc7ac\ubc30\uc5f4\ub41c \ubb34\ud55c\uae09\uc218<\/h2>\n<p>\uc591\uc758 \uc815\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc5d0\uc11c \uc790\uae30 \uc790\uc2e0\uc73c\ub85c \uac00\ub294 \uc77c\ub300\uc77c \ub300\uc751 \\(r\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uace0<br \/>\n\\[<br \/>\nr_n=r(n)<br \/>\n\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uc790. \uc218\uc5f4<br \/>\n\\[<br \/>\na_{r_1},\\,a_{r_2},\\,a_{r_3},\\,\\ldots<br \/>\n\\]<br \/>\n\uc744 \\(\\left\\{a_n\\right\\}\\)\uc758 <span class=\"defined\">\uc7ac\ubc30\uc5f4<\/span>(rearrangement)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \uc7ac\ubc30\uc5f4\uc5d0\uc11c\ub294 \uc6d0\ub798 \uc218\uc5f4\uc758 \uac01 \ud56d\uc744 \uc815\ud655\ud788 \ud55c \ubc88\uc529 \uc0ac\uc6a9\ud558\ub418 \uadf8 \uc21c\uc11c\ub9cc \ubc14\uafbc\ub2e4.<\/p>\n<div class=\"lemma margintop2\">\n<p><span class=\"theorem\">\ubcf4\uc870\uc815\ub9ac. (\uc74c\uc774 \uc544\ub2cc \ud56d\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uae09\uc218\uc758 \uc7ac\ubc30\uc5f4)<\/span><\/p>\n<p>\\(a_n\\ge0\\)\uc774\uace0<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_n=S<br \/>\n\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \\(\\left\\{a_{r_n}\\right\\}\\)\uc774 \\(\\left\\{a_n\\right\\}\\)\uc758 \uc7ac\ubc30\uc5f4\uc774\uba74<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_{r_n}=S.<br \/>\n\\]\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\uc7ac\ubc30\uc5f4\ub41c \uae09\uc218\uc758 \\(N\\)\ubc88\uc9f8 \ubd80\ubd84\ud569\uc744<br \/>\n\\[<br \/>\nT_N<br \/>\n=<br \/>\n\\sum_{k=1}^{N}a_{r_k}<br \/>\n\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \\(r_1,\\ldots,r_N\\) \uac00\uc6b4\ub370 \uac00\uc7a5 \ud070 \uac12\uc744 \\(m\\)\uc774\ub77c\uace0 \ud558\uba74 \ubaa8\ub4e0 \ud56d\uc774 \uc74c\uc774 \uc544\ub2c8\ubbc0\ub85c<br \/>\n\\[<br \/>\nT_N<br \/>\n\\le<br \/>\n\\sum_{k=1}^{m}a_k<br \/>\n\\le<br \/>\nS.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(\\left\\{T_N\\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud558\uba74\uc11c \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c \uc5b4\ub5a4 \uc2e4\uc218 \\(T\\)\uc5d0 \uc218\ub834\ud558\uace0<br \/>\n\\[<br \/>\nT\\le S.<br \/>\n\\]\n<\/p>\n<p>\ubc18\ub300\ub85c \uc591\uc758 \uc815\uc218 \\(m\\)\uc744 \uace0\uc815\ud558\uc790. \\(r\\)\uac00 \uc77c\ub300\uc77c \ub300\uc751\uc774\ubbc0\ub85c \\(1,2,\\ldots,m\\)\uc740 \uac01\uac01 \uc7ac\ubc30\uc5f4\ub41c \uc218\uc5f4\uc758 \uc5b4\ub290 \uc704\uce58\uc5d4\uac00 \ub098\ud0c0\ub09c\ub2e4. \ub530\ub77c\uc11c \ucda9\ubd84\ud788 \ud070 \\(N\\)\uc744 \ud0dd\ud558\uba74<br \/>\n\\[<br \/>\n\\{1,2,\\ldots,m\\}<br \/>\n\\subseteq<br \/>\n\\{r_1,r_2,\\ldots,r_N\\}.<br \/>\n\\]<br \/>\n\ubaa8\ub4e0 \ud56d\uc774 \uc74c\uc774 \uc544\ub2c8\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\sum_{k=1}^{m}a_k<br \/>\n\\le<br \/>\nT_N<br \/>\n\\le<br \/>\nT.<br \/>\n\\]<br \/>\n\\(m\\rightarrow\\infty\\)\ub85c \ubcf4\ub0b4\uba74<br \/>\n\\[<br \/>\nS\\le T.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c<br \/>\n\\[<br \/>\nS=T.<br \/>\n\\]<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 2.3.3. (\uc808\ub300\uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\uc758 \uc7ac\ubc30\uc5f4)<\/span><\/p>\n<p>\ubb34\ud55c\uae09\uc218<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_n<br \/>\n\\]<br \/>\n\uc774 \uc808\ub300\uc218\ub834\ud558\uace0 \\(\\left\\{a_{r_n}\\right\\}\\)\uc774 \\(\\left\\{a_n\\right\\}\\)\uc758 \uc7ac\ubc30\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_{r_n}<br \/>\n\\]<br \/>\n\ub3c4 \uc808\ub300\uc218\ub834\ud558\uba70<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_{r_n}<br \/>\n=<br \/>\n\\sum_{n=1}^{\\infty}a_n<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\uae09\uc218<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}|a_n|<br \/>\n\\]<br \/>\n\uc740 \uc74c\uc774 \uc544\ub2cc \ud56d\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc218\ub834\uae09\uc218\uc774\uace0<br \/>\n\\[<br \/>\n\\left\\{|a_{r_n}|\\right\\}<br \/>\n\\]<br \/>\n\uc740 \\(\\left\\{|a_n|\\right\\}\\)\uc758 \uc7ac\ubc30\uc5f4\uc774\ub2e4. \ub530\ub77c\uc11c \uc55e\uc758 \ubcf4\uc870\uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}|a_{r_n}|<br \/>\n=<br \/>\n\\sum_{n=1}^{\\infty}|a_n|<br \/>\n<\n\\infty.\n\\]\n\uadf8\ub7ec\ubbc0\ub85c \uc7ac\ubc30\uc5f4\ub41c \uae09\uc218\ub3c4 \uc808\ub300\uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\ub610\ud55c \ubcf4\uc870\uc815\ub9ac 2.3.1\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_n^+,<br \/>\n\\qquad<br \/>\n\\sum_{n=1}^{\\infty}a_n^-<br \/>\n\\]<br \/>\n\uac00 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4. \\(\\left\\{a_{r_n}^+\\right\\}\\)\uacfc \\(\\left\\{a_{r_n}^-\\right\\}\\)\uc740 \uac01\uac01 \uc774 \ub450 \uc218\uc5f4\uc758 \uc7ac\ubc30\uc5f4\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_{r_n}^+<br \/>\n=<br \/>\n\\sum_{n=1}^{\\infty}a_n^+<br \/>\n\\]<br \/>\n\uc774\uace0<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_{r_n}^-<br \/>\n=<br \/>\n\\sum_{n=1}^{\\infty}a_n^-.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c<br \/>\n\\[<br \/>\n\\begin{aligned}<br \/>\n\\sum_{n=1}^{\\infty}a_{r_n}<br \/>\n&#038;=<br \/>\n\\sum_{n=1}^{\\infty}a_{r_n}^+<br \/>\n&#8211;<br \/>\n\\sum_{n=1}^{\\infty}a_{r_n}^-\\\\<br \/>\n&#038;=<br \/>\n\\sum_{n=1}^{\\infty}a_n^+<br \/>\n&#8211;<br \/>\n\\sum_{n=1}^{\\infty}a_n^-\\\\<br \/>\n&#038;=<br \/>\n\\sum_{n=1}^{\\infty}a_n.<br \/>\n\\end{aligned}<br \/>\n\\]<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc808\ub300\uc218\ub834\ud558\ub294 \uae09\uc218\uc5d0\uc11c\ub294 \ud56d\uc758 \uc21c\uc11c\ub97c \uc5b4\ub5bb\uac8c \ubc14\uafb8\uc5b4\ub3c4 \ud569\uc774 \ubcc0\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ub098 \uc870\uac74\uc218\ub834\ud558\ub294 \uae09\uc218\uc5d0\uc11c\ub294 \uc0c1\ud669\uc774 \uc804\ud600 \ub2e4\ub974\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 2.3.4. (\ub9ac\ub9cc\uc758 \uc7ac\ubc30\uc5f4 \uc815\ub9ac)<\/span><\/p>\n<p>\ubb34\ud55c\uae09\uc218<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}a_n<br \/>\n\\]<br \/>\n\uc774 \uc870\uac74\uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uc784\uc758\uc758 \uc2e4\uc218 \\(A\\)\uc5d0 \ub300\ud558\uc5ec \uc6d0\ub798 \uae09\uc218\uc758 \ud56d\uc744 \uc7ac\ubc30\uc5f4\ud558\uc5ec \ud569\uc774 \\(A\\)\uc778 \uae09\uc218\ub97c \ub9cc\ub4e4 \uc218 \uc788\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\uba3c\uc800 \uc870\uac74\uc218\ub834\ud558\ub294 \uae09\uc218\uc758 \uc591\uc758 \ud56d\ub4e4\uc758 \ud569\uacfc \uc74c\uc758 \ud56d\ub4e4\uc758 \uc808\ub313\uac12\uc758 \ud569\uc774 \ubaa8\ub450 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud568\uc744 \ubcf4\uc774\uc790.<\/p>\n<p>\n\\[<br \/>\nS_N=\\sum_{n=1}^{N}a_n,<br \/>\n\\qquad<br \/>\nA_N=\\sum_{n=1}^{N}|a_n|<br \/>\n\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uc790. \uc6d0\ub798 \uae09\uc218\ub294 \uc218\ub834\ud558\ubbc0\ub85c \uc5b4\ub5a4 \uc2e4\uc218 \\(S\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nS_N\\rightarrow S.<br \/>\n\\]<br \/>\n\ubc18\uba74 \uc808\ub300\uc218\ub834\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c \uc74c\uc774 \uc544\ub2cc \ud56d\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uae09\uc218<br \/>\n\\[<br \/>\n\\sum_{n=1}^{\\infty}|a_n|<br \/>\n\\]<br \/>\n\uc740 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\nA_N\\rightarrow\\infty.<br \/>\n\\]\n<\/p>\n<p>\ub610\ud55c<br \/>\n\\[<br \/>\nP_N<br \/>\n=<br \/>\n\\sum_{n=1}^{N}a_n^+<br \/>\n=<br \/>\n\\frac{A_N+S_N}{2}<br \/>\n\\]<br \/>\n\uc774\uace0<br \/>\n\\[<br \/>\nQ_N<br \/>\n=<br \/>\n\\sum_{n=1}^{N}a_n^-<br \/>\n=<br \/>\n\\frac{A_N-S_N}{2}.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c<br \/>\n\\[<br \/>\nP_N\\rightarrow\\infty,<br \/>\n\\qquad<br \/>\nQ_N\\rightarrow\\infty.<br \/>\n\\]<br \/>\n\uc989 \uc591\uc758 \ud56d\ub4e4\uc758 \ud569\uc740 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uace0, \uc74c\uc758 \ud56d\ub4e4\uc758 \uc808\ub313\uac12\uc758 \ud569\ub3c4 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n<p>\uc6d0\ub798 \uae09\uc218\uc5d0 \ub098\ud0c0\ub098\ub294 \uc591\uc758 \ud56d\ub4e4\uc744 \uc6d0\ub798\uc758 \uc21c\uc11c\ub300\ub85c<br \/>\n\\[<br \/>\np_1,p_2,p_3,\\ldots<br \/>\n\\]<br \/>\n\ub77c\uace0 \ud558\uace0, \uc74c\uc758 \ud56d\ub4e4\uc744<br \/>\n\\[<br \/>\n-q_1,-q_2,-q_3,\\ldots<br \/>\n\\qquad<br \/>\n(q_j>0)<br \/>\n\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[<br \/>\n\\sum_{j=1}^{\\infty}p_j=\\infty,<br \/>\n\\qquad<br \/>\n\\sum_{j=1}^{\\infty}q_j=\\infty.<br \/>\n\\]<br \/>\n\ub610\ud55c \uc6d0\ub798 \uae09\uc218\uac00 \uc218\ub834\ud558\ubbc0\ub85c \\(a_n\\rightarrow0\\)\uc774\uace0, \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\np_j\\rightarrow0,<br \/>\n\\qquad<br \/>\nq_j\\rightarrow0.<br \/>\n\\]\n<\/p>\n<p>\uc774\uc81c \ub2e4\uc74c\uacfc \uac19\uc774 \ud56d\uc744 \uc7ac\ubc30\uc5f4\ud55c\ub2e4. \uc0ac\uc6a9\ud558\uc9c0 \uc54a\uc740 \uc591\uc758 \ud56d\ub4e4\uc744 \uc6d0\ub798 \uc21c\uc11c\ub300\ub85c \ud558\ub098\uc529 \ub354\ud558\uc5ec \ubd80\ubd84\ud569\uc774 \\(A\\)\ubcf4\ub2e4 \ucee4\uc9c8 \ub54c\uae4c\uc9c0 \uacc4\uc18d\ud55c\ub2e4. \uadf8\ub2e4\uc74c \uc0ac\uc6a9\ud558\uc9c0 \uc54a\uc740 \uc74c\uc758 \ud56d\ub4e4\uc744 \uc6d0\ub798 \uc21c\uc11c\ub300\ub85c \ud558\ub098\uc529 \ub354\ud558\uc5ec \ubd80\ubd84\ud569\uc774 \\(A\\)\ubcf4\ub2e4 \uc791\uc544\uc9c8 \ub54c\uae4c\uc9c0 \uacc4\uc18d\ud55c\ub2e4. \ub2e4\uc2dc \uc591\uc758 \ud56d\uc744 \ub354\ud558\uc5ec \\(A\\)\ub97c \ub118\uace0, \ub2e4\uc2dc \uc74c\uc758 \ud56d\uc744 \ub354\ud558\uc5ec \\(A\\) \uc544\ub798\ub85c \ub0b4\ub824\uac00\ub294 \uacfc\uc815\uc744 \ubc18\ubcf5\ud55c\ub2e4.<\/p>\n<p>\uc591\uc758 \ud56d\uc744 \ub354\ud55c \uc9c1\ud6c4\uc758 \ubd80\ubd84\ud569\ub4e4\uc744 \\(U_1,U_2,\\ldots\\), \uc74c\uc758 \ud56d\uc744 \ub354\ud55c \uc9c1\ud6c4\uc758 \ubd80\ubd84\ud569\ub4e4\uc744 \\(V_1,V_2,\\ldots\\)\ub77c\uace0 \ud558\uc790. \uac01 \ub2e8\uacc4\uc5d0\uc11c \\(A\\)\ub97c \ucc98\uc74c \ub118\uc5b4\uc11c\ub294 \uc21c\uac04 \uba48\ucd94\ubbc0\ub85c, \ucda9\ubd84\ud788 \ud070 \\(j\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n0<U_j-A\\le p_{i_j}\n\\]\n\uc778 \uc5b4\ub5a4 \\(i_j\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c\n\\[\n0<A-V_j\\le q_{\\ell_j}\n\\]\n\uc778 \uc5b4\ub5a4 \\(\\ell_j\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<p>\uc7ac\ubc30\uc5f4 \uacfc\uc815\uc5d0\uc11c \uc591\uc758 \ud56d\uacfc \uc74c\uc758 \ud56d\uc744 \uacc4\uc18d \uc0c8\ub85c \uc0ac\uc6a9\ud558\ubbc0\ub85c<br \/>\n\\[<br \/>\ni_j\\rightarrow\\infty,<br \/>\n\\qquad<br \/>\n\\ell_j\\rightarrow\\infty.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c<br \/>\n\\[<br \/>\np_{i_j}\\rightarrow0,<br \/>\n\\qquad<br \/>\nq_{\\ell_j}\\rightarrow0,<br \/>\n\\]<br \/>\n\uc774\uace0<br \/>\n\\[<br \/>\nU_j\\rightarrow A,<br \/>\n\\qquad<br \/>\nV_j\\rightarrow A.<br \/>\n\\]\n<\/p>\n<p>\uc591\uc758 \ud56d\uc744 \ub354\ud558\ub294 \ub3d9\uc548\uc758 \ubaa8\ub4e0 \uc911\uac04 \ubd80\ubd84\ud569\uc740 \ubc14\ub85c \uc55e\uc758 \\(V_j\\)\uc640 \ub2e4\uc74c \\(U_{j+1}\\) \uc0ac\uc774\uc5d0 \uc788\uace0, \uc74c\uc758 \ud56d\uc744 \ub354\ud558\ub294 \ub3d9\uc548\uc758 \ubaa8\ub4e0 \uc911\uac04 \ubd80\ubd84\ud569\uc740 \\(U_j\\)\uc640 \\(V_j\\) \uc0ac\uc774\uc5d0 \uc788\ub2e4. \ub530\ub77c\uc11c \uc7ac\ubc30\uc5f4\ub41c \uae09\uc218\uc758 \ubaa8\ub4e0 \ubd80\ubd84\ud569\uc740 \\(A\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\uc774 \uacfc\uc815\uc740 \uc591\uc758 \ud56d\uacfc \uc74c\uc758 \ud56d\uc744 \uac01\uac01 \uc6d0\ub798 \uc21c\uc11c\ub300\ub85c \ubaa8\ub450 \uc0ac\uc6a9\ud55c\ub2e4. \uc6d0\ub798 \uae09\uc218\uc5d0 \\(0\\)\uc778 \ud56d\uc774 \uc788\ub2e4\uba74 \uadf8 \ud56d\ub4e4\ub3c4 \uacfc\uc815 \uc911\uc5d0 \ucc28\ub840\ub85c \uc0bd\uc785\ud558\uba74 \ub41c\ub2e4. \\(0\\)\uc778 \ud56d\uc740 \ubd80\ubd84\ud569\uc758 \uac12\uc744 \ubc14\uafb8\uc9c0 \uc54a\ub294\ub2e4. \ub530\ub77c\uc11c \uc774\ub807\uac8c \uc5bb\uc740 \uae09\uc218\ub294 \uc6d0\ub798 \uae09\uc218\uc758 \uc2e4\uc81c \uc7ac\ubc30\uc5f4\uc774\uace0 \uadf8 \ud569\uc740 \\(A\\)\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2.3.2.<\/span><br \/>\n\ub2e4\uc74c \ub450 \uae09\uc218\ub97c \uc0dd\uac01\ud558\uc790.<br \/>\n\\[<br \/>\n\\begin{aligned}<br \/>\n&#038;1-\\frac12+\\frac13-\\frac14+\\frac15-\\frac16+\\cdots,<br \/>\n\\\\[5pt]<br \/>\n&#038;1-\\frac12-\\frac14+\\frac13-\\frac16-\\frac18<br \/>\n+\\frac15-\\frac1{10}-\\frac1{12}+\\cdots.<br \/>\n\\end{aligned}<br \/>\n\\]<br \/>\n\ub458\uc9f8 \uae09\uc218\ub294 \uccab\uc9f8 \uae09\uc218\uc758 \ud56d\uc744 \uc7ac\ubc30\uc5f4\ud55c \uae09\uc218\uc774\ub2e4.<\/p>\n<p>\uccab\uc9f8 \uae09\uc218\uc640 \ub458\uc9f8 \uae09\uc218\uc758 \ubd80\ubd84\ud569\uc744 \uac01\uac01 \\(S_n,T_n\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[<br \/>\n\\begin{aligned}<br \/>\nT_{3n}<br \/>\n&#038;=<br \/>\n\\sum_{k=1}^{n}<br \/>\n\\left(<br \/>\n\\frac1{2k-1}<br \/>\n&#8211;<br \/>\n\\frac1{4k-2}<br \/>\n&#8211;<br \/>\n\\frac1{4k}<br \/>\n\\right)<br \/>\n\\\\<br \/>\n&#038;=<br \/>\n\\frac12<br \/>\n\\sum_{k=1}^{n}<br \/>\n\\left(<br \/>\n\\frac1{2k-1}<br \/>\n&#8211;<br \/>\n\\frac1{2k}<br \/>\n\\right)<br \/>\n\\\\<br \/>\n&#038;=<br \/>\n\\frac12S_{2n}.<br \/>\n\\end{aligned}<br \/>\n\\]\n<\/p>\n<p>\uc774 \uad00\uacc4\ub97c \uc774\uc6a9\ud558\uc5ec \uccab\uc9f8 \uae09\uc218\uac00 \uc808\ub300\uc218\ub834\ud560 \uc218 \uc5c6\uc74c\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4. \uccab\uc9f8 \uae09\uc218\uac00 \uc808\ub300\uc218\ub834\ud558\uace0 \uadf8 \ud569\uc774 \\(S\\)\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \uc815\ub9ac 2.3.3\uc5d0 \uc758\ud558\uc5ec \ub458\uc9f8 \uae09\uc218\ub3c4 \\(S\\)\uc5d0 \uc218\ub834\ud574\uc57c \ud55c\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\nT_{3n}\\rightarrow S.<br \/>\n\\]<br \/>\n\ud55c\ud3b8<br \/>\n\\[<br \/>\nS_{2n}\\rightarrow S<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\nT_{3n}<br \/>\n=<br \/>\n\\frac12S_{2n}<br \/>\n\\rightarrow<br \/>\n\\frac12S.<br \/>\n\\]<br \/>\n\uadf9\ud55c\uc758 \uc720\uc77c\uc131\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[<br \/>\nS=\\frac12S,<br \/>\n\\]<br \/>\n\uc989 \\(S=0\\)\uc774\uc5b4\uc57c \ud55c\ub2e4.<\/p>\n<p>\uadf8\ub7ec\ub098<br \/>\n\\[<br \/>\nS_{2n}<br \/>\n=<br \/>\n\\sum_{k=1}^{n}<br \/>\n\\left(<br \/>\n\\frac1{2k-1}-\\frac1{2k}<br \/>\n\\right)<br \/>\n\\ge<br \/>\n\\frac12<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \\(S\\ge\\frac12\\)\uc774\ub2e4. \uc774\ub294 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c \uccab\uc9f8 \uae09\uc218\ub294 \uc808\ub300\uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \ub2e4\uc74c \uc808\uc5d0\uc11c \uc774 \uae09\uc218 \uc790\uccb4\ub294 \uc218\ub834\ud568\uc744 \ubcf4\uc77c \uac83\uc774\ubbc0\ub85c, \uc774 \uae09\uc218\ub294 \uc870\uac74\uc218\ub834\ud558\ub294 \uae09\uc218\uc758 \ub300\ud45c\uc801\uc778 \uc608\uac00 \ub41c\ub2e4.<\/p>\n<\/div>\n<p><!-- ##################################################################### --><br \/>\n<!--\n\n\n<h2 class=\"itc_h2\">\uc81c\ubaa9<\/h2>\n\n\n\n\n\n<p>.<\/p>\n\n\n\n\n\n<p>.<\/p>\n\n\n--><\/p>\n<p><!--\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\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<\/div>\n\n\n\n\n########\n\n\u201c\u201d\n\u2018\u2019\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<h2 class=\"itc_h2\"><\/h2>\n\n\n\n--><\/p>\n<p><!--\n\n\n<div style=\"display: none; visibility: hidden;\">\n\\[\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}\n\\newcommand{\\proj}{{\\operatorname{proj}}}\n\\newcommand{\\adj}{{\\operatorname{adj}}}\n\\]\n<\/div>\n\n\n--><\/p>\n<div class=\"box itc_prev_next_box\">\n<ul class=\"itc_ul\">\n<li class=\"itc_li_prev\">\uc55e\uc758 \uae00 : <a href=\"..\/series-of-nonnegative-terms\">\uc591\ud56d\uae09\uc218<\/a><\/li>\n<li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/alternating-series\">\uad50\ub300\uae09\uc218<\/a><\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 2\uc7a5 3\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \\(\\left\\{a_n\\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \ubb34\ud55c\uae09\uc218\uc758 \uc808\ub300\uc218\ub834\uacfc \uc870\uac74\uc218\ub834\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4. \ubb34\ud55c\uae09\uc218 \\( \\sum_{n=1}^{\\infty}|a_n| \\) \uc774 \uc218\ub834\ud558\uba74 \ubb34\ud55c\uae09\uc218 \\( \\sum_{n=1}^{\\infty}a_n \\) \uc774 \uc808\ub300\uc218\ub834\ud55c\ub2e4(converge absolutely)\uace0 \ub9d0\ud55c\ub2e4. \ubb34\ud55c\uae09\uc218 \\( \\sum_{n=1}^{\\infty}a_n \\) \uc774 \uc218\ub834\ud558\uc9c0\ub9cc \uc808\ub300\uc218\ub834\ud558\uc9c0 \uc54a\uc73c\uba74 \uc774 \uae09\uc218\uac00 \uc870\uac74\uc218\ub834\ud55c\ub2e4(converge conditionally)\uace0 \ub9d0\ud55c\ub2e4. \uc808\ub300\uc218\ub834\uacfc \uc870\uac74\uc218\ub834\uc758 \uad00\uacc4 \\(\\left\\{a_n\\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \\(a_n\\)\uc758 \uc591\uc758 \ubd80\ubd84(positive part) \\(a_n^+\\)\uacfc \uc74c\uc758 \ubd80\ubd84(negative part) \\(a_n^-\\)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":203,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-6686","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6686","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"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=6686"}],"version-history":[{"count":16,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6686\/revisions"}],"predecessor-version":[{"id":10141,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6686\/revisions\/10141"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6620"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6686"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}