{"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":"2021-09-30T18:30:26","modified_gmt":"2021-09-30T09:30:26","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>\ub9cc\uc57d \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} \\left\\lvert a_n \\right\\rvert\\)\uc774 \uc218\ub834\ud558\uba74 \u201c\ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 <span class=\"defined\">\uc808\ub300\uc218\ub834<\/span>\ud55c\ub2e4(converge absolutely)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<li>\ub9cc\uc57d \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud558\uc9c0\ub9cc \uc808\ub300\uc218\ub834\ud558\uc9c0\ub294 \uc54a\uc73c\uba74 \u201c\ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 <span class=\"defined\">\uc870\uac74\uc218\ub834<\/span>\ud55c\ub2e4(converge conditionally)\u201d\ub77c\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\uc77c \ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\na_n^+ &#038;=<br \/>\n\\begin{cases}<br \/>\na_n &#038; \\quad \\text{if} \\,\\, a_n \\ge 0,\\\\<br \/>\n0 &#038; \\quad \\text{if} \\,\\, a_n < 0;\n\\end{cases}\n\\\\[6pt]\na_n^- &#038;=\n\\begin{cases}\n-a_n &#038; \\quad \\text{if} \\,\\, a_n \\le 0,\\\\\n0 &#038; \\quad \\text{if} \\,\\, a_n > 0.<br \/>\n\\end{cases}<br \/>\n\\end{align}\\]<br \/>\n\uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[a_n = a_n^+ &#8211; a_n^- \\quad\\text{and}\\quad \\left\\lvert a_n \\right\\rvert = a_n^+ + a_n^-\\]<br \/>\n\uc774\ub2e4. \ub610\ud55c<br \/>\n\\[a_n^+ \\ge 0 \\quad\\text{and}\\quad a_n^- \\ge 0\\]<br \/>\n\uc774\uba70,<br \/>\n\\[a_n^+ \\le \\left\\lvert a_n \\right\\rvert \\quad\\text{and}\\quad a_n^- \\le \\left\\lvert a_n \\right\\rvert\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c, \ub9cc\uc57d \\(\\sum_{n=1}^{\\infty} \\left\\lvert a_n \\right\\rvert\\)\uc774 \uc218\ub834\ud558\uba74 \\(\\sum_{n=1}^{\\infty} a_n ^+\\)\uc640 \\(\\sum_{n=1}^{\\infty} a_n^-\\)\uac00 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\uc5ed\uc73c\ub85c \\(\\sum_{n=1}^{\\infty} a_n ^+\\)\uc640 \\(\\sum_{n=1}^{\\infty} a_n^-\\)\uac00 \ubaa8\ub450 \uc218\ub834\ud558\uba74 \\(\\sum_{n=1}^{\\infty} \\left\\lvert a_n \\right\\rvert\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\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 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\sum_{n=1}^{\\infty} a_n ^+\\)\uc640 \\(\\sum_{n=1}^{\\infty} a_n^-\\)\uac00 \ubaa8\ub450 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[a_n^+ \\le \\left\\lvert a_n \\right\\rvert \\quad\\text{and}\\quad a_n^- \\le \\left\\lvert a_n \\right\\rvert\\]<br \/>\n\uc774\ubbc0\ub85c \\(\\sum_{n=1}^{\\infty} a_n^+\\)\uc640 \\(\\sum_{n=1}^{\\infty} a_n^-\\)\uac00 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7f0\ub370<br \/>\n\\[\\sum_{k=1}^n a_k = \\sum_{k=1}^n a_k^+ &#8211; \\sum_{k=1}^n a_k ^-\\]<br \/>\n\uc774\ubbc0\ub85c \\(\\sum_{n=1}^{\\infty}a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 2.3.2.<\/span><br \/>\n\ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_n \\)\uc774 \uc808\ub300\uc218\ub834\ud558\uba74, \uc774 \ubb34\ud55c\uae09\uc218\ub294 \uc218\ub834\ud55c\ub2e4.\n<\/p>\n<\/div>\n<p><span class=\"definition\">\ucc38\uace0.<\/span><br \/>\n\uc704 \uc815\ub9ac\uc758 \uc5ed\uc740 \uc77c\ubc18\uc801\uc73c\ub85c \ucc38\uc774 \uc544\ub2c8\ub2e4. \ub2e4\uc74c \ubb34\ud55c\uae09\uc218\ub97c \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[\\sum_{n=1}^{\\infty} \\frac{(-1)^n}{n}.\\]<br \/>\n\uc774 \ubb34\ud55c\uae09\uc218\ub294 \uad50\ub300\uae09\uc218 \ud310\uc815\ubc95(2.4\uc808\uc744 \ucc38\uace0\ud558\ub77c)\uc5d0 \uc758\ud558\uc5ec \uc218\ub834\ud558\uc9c0\ub9cc, \uc808\ub300\uc218\ub834\ud558\uc9c0\ub294 \uc54a\ub294\ub2e4.<br \/>\n<span class=\"qee\"><\/span><\/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 \\(x\\)\uc758 \uac12\uc758 \ubc94\uc704\ub97c \uad6c\ud574 \ubcf4\uc790.<br \/>\n\\[1 + 2x + x^2 + 2x^3 + x^4 + 2x^5 + \\cdots .\\]<br \/>\n\uc608\uc81c 2.2.7\uc5d0\uc11c \\(0 \\le x < 1\\)\uc77c \ub54c \uc774 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud558\uace0 \\(x \\ge 1\\)\uc77c \ub54c \uc774 \ubb34\ud55c\uae09\uc218\uac00 \ubc1c\uc0b0\ud568\uc744 \ubcf4\uc600\ub2e4. \ub530\ub77c\uc11c \\(-1 < x < 0\\)\uc77c \ub54c \ubb34\ud55c\uae09\uc218\n\\[1 + 2 \\lvert x \\rvert + \\lvert x \\rvert^2 + 2 \\lvert x \\rvert^3 + \\lvert x \\rvert^4 + 2 \\lvert x \\rvert^5 + \\cdots\\]\n\uc774 \uc218\ub834\ud55c\ub2e4. \uc989 \\(-1 < x < 0\\)\uc77c \ub54c \ubb38\uc81c\uc758 \ubb34\ud55c\uae09\uc218\uac00 \uc808\ub300\uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(-1 < x < 1\\)\uc77c \ub54c \ubb38\uc81c\uc758 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\\(x\\le -1\\)\uc774\uac70\ub098 \\(x \\ge 1\\)\uc77c \ub54c\ub294 \ubb34\ud55c\uae09\uc218\uc758 \ud56d\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c \ubb38\uc81c\uc758 \ubb34\ud55c\uae09\uc218\uac00 \ubc1c\uc0b0\ud55c\ub2e4.\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc7ac\ubc30\uc5f4\ub41c \ubb34\ud55c\uae09\uc218<\/h2>\n<p>\uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc774\uba70<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n = S\\]<br \/>\n\ub85c\uc11c \\(\\left\\{a_n\\right\\}\\)\uc758 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\left\\{ a_{r_n}\\right\\}\\)\uc774 \\(\\left\\{ a_n \\right\\}\\)\uc744 \uc7ac\ubc30\uc5f4\ud55c \uc218\uc5f4(\ud56d\uc758 \uc21c\uc11c\ub97c \ubc14\uafbc \uc218\uc5f4)\uc774\ub77c\uace0 \ud558\uc790. \uc74c\uc774 \uc544\ub2cc \ud56d\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\uc758 \ubd80\ubd84\ud569\uc740 \ubb34\ud55c\uae09\uc218\uc758 \ud569\uc744 \ub118\uc744 \uc218 \uc5c6\uc73c\ubbc0\ub85c<br \/>\n\\[\\sum_{k=1}^n a_{r_k} \\le S\\]<br \/>\n\uc774\ub2e4. \uc989 \\(\\sum_{k=1}^n a_{r_k}\\)\ub294 \uc720\uacc4\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\sum_{k=1}^{\\infty}a_{r_k}\\)\ub294 \uc801\ub2f9\ud55c \uc2e4\uc218 \\(T\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uc774\ub54c \\(T\\le S\\)\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\uad00\uc810\uc744 \ubc14\uafb8\uc5b4 \ubcf4\uba74 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \\(\\sum_{k=1}^{\\infty}a_{r_k}\\)\uc758 \ud56d\uc758 \uc21c\uc11c\ub97c \ubc14\uafbc \ubb34\ud55c\uae09\uc218\uc774\ubbc0\ub85c \\(\\sum_{k=1}^n a_k \\le T\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(S\\le T\\)\uc774\ub2e4.<\/p>\n<p>\uc774\ub85c\uc368 \ub4f1\uc2dd \\(T=S\\)\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<p>\uc774\uc81c \\(\\left\\{ a_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc774\ub77c\ub294 \uc870\uac74\uc744 \uc81c\uc678\ud558\uc790. \ub300\uc2e0 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \\(\\left\\{ a_{r_n}\\right\\}\\)\uc774 \\(\\left\\{ a_n \\right\\}\\)\uc744 \uc7ac\ubc30\uc5f4\ud55c \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\left\\{a_{r_n}^+\\right\\}\\)\uc640 \\(\\left\\{a_{r_n}^-\\right\\}\\)\ub294 \uac01\uac01 \\(\\left\\{ a_n ^+ \\right\\}\\)\uc640 \\(\\left\\{ a_n ^- \\right\\}\\)\ub97c \uc7ac\ubc30\uc5f4\ud55c \uc218\uc5f4\uc774\ub2e4. \uc5ec\uae30\uc5d0 \ubcf4\uc870\uc815\ub9ac 2.3.1\uc744 \uc0ac\uc6a9\ud558\uba74 \ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n = \\sum_{n=1}^{\\infty} a_n^+ &#8211; \\sum_{n=1}^{\\infty} a_n^- = \\sum_{n=1}^{\\infty}  a_{r_n}^+ &#8211; \\sum_{n=1}^{\\infty} a_{r_n}^- = \\sum_{n=1}^{\\infty} a_{r_n}.\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\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 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \\(\\left\\{ a_{r_n}\\right\\}\\)\uc774 \\(\\left\\{a_n\\right\\}\\)\uc744 \uc7ac\ubc30\uc5f4\ud55c \uc218\uc5f4\uc774\ub77c\uba74 \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty}a_{r_n}\\)\ub3c4 \uc808\ub300\uc218\ub834\ud558\uba70,<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n = \\sum_{n=1}^{\\infty} a_{r_n}\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n<p>\uc774\ubc88\uc5d0\ub294 \uc870\uac74\uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\ub97c \uc7ac\ubc30\uc5f4\ud55c \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \ubb34\ud55c\uae09\uc218<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n\\]<br \/>\n\uc774 \uc870\uac74\uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub450 \ubb34\ud55c\uae09\uc218<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n^+ ,\\quad \\sum_{n=1}^{\\infty} a_n^-\\]<br \/>\n\ub294 \ubaa8\ub450 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n<p>\\(A\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc9c4 \uc218\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\\(\\left\\{ a_n ^+ \\right\\}\\)\uc758 \ud56d\uc744 \uc55e\uc5d0\uc11c\ubd80\ud130 \ucc28\ub840\ub85c \ub354\ud574\uac00\ub2e4 \ubcf4\uba74 \uadf8 \ud569 \\(P_1\\)\uc774 \\(A\\)\ubcf4\ub2e4 \ucee4\uc9c0\ub294 \uc21c\uac04\uc774 \uc628\ub2e4. \uc989 \\[P_1 > A\\]\uc774\ub2e4. \uc5ec\uae30\uc11c \uba48\ucd98 \ud6c4, \ub2e4\uc2dc \\(\\left\\{ -a_n ^- \\right\\}\\)\uc758 \ud56d\uc744 \uc55e\uc5d0\uc11c\ubd80\ud130 \ucc28\ub840\ub85c \ub354\ud574\uac00\ub2e4 \ubcf4\uba74 \uadf8 \ud569 \\(Q_1\\)\uc774 \\(A-P_1\\)\ubcf4\ub2e4 \uc791\uc544\uc9c0\ub294 \uc21c\uac04\uc774 \uc628\ub2e4. \uc989 \\[P_1 + Q_1 < A\\]\uc774\ub2e4. \uc5ec\uae30\uc11c \uba48\ucd98 \ud6c4, \ub2e4\uc2dc \\(\\left\\{ a_n^+ \\right\\}\\)\uc758 \ub0a8\uc740 \ud56d\uc744 \uc55e\uc5d0\uc11c\ubd80\ud130 \ucc28\ub840\ub85c \ub354\ud574\uac00\ub2e4 \ubcf4\uba74 \uadf8 \ud569 \\(P_2\\)\uac00 \\(A-P_1 - Q_1\\)\ubcf4\ub2e4 \ucee4\uc9c0\ub294 \uc21c\uac04\uc774 \uc628\ub2e4. \uc989 \\[P_1 + Q_1 + P_2 > A\\]\uc774\ub2e4. \uc5ec\uae30\uc11c \uba48\ucd98 \ud6c4, \ub2e4\uc2dc \\(\\left\\{ -a_n^- \\right\\}\\)\uc758 \ub0a8\uc740 \ud56d\uc744 \uc55e\uc5d0\uc11c\ubd80\ud130 \ucc28\ub840\ub85c \ub354\ud574\uac00\ub2e4 \ubcf4\uba74 \uadf8 \ud569 \\(Q_2\\)\uac00 \\(A &#8211; P_1 &#8211; Q_1 &#8211; P_2\\)\ubcf4\ub2e4 \uc791\uc544\uc9c0\ub294 \uc21c\uac04\uc774 \uc628\ub2e4. \uc989 \\[P_1 +Q_1 +P_2 + Q_1 < A\\]\uc774\ub2e4.<\/p>\n<p>\uc774 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uba74 \\(A\\)\uc5d0 \uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218<br \/>\n\\[P_1 + Q_1 + P_2 + Q_2 + P_3 + Q_3 + \\cdots\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4. \uc774 \ubb34\ud55c\uae09\uc218\ub294 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc758 \ud56d\uc744 \uc7ac\ubc30\uc5f4\ud55c \ubb34\ud55c\uae09\uc218\uc774\ub2e4.<\/p>\n<p>\uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\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 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc870\uac74\uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(A\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc9c4 \uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc758 \ud56d\uc744 \uc7ac\ubc30\uc5f4\ud558\uc5ec \\(A\\)\uc5d0 \uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_{r_n}\\)\uc744 \ub9cc\ub4e4 \uc218 \uc788\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2.3.2.<\/span><br \/>\n\ub2e4\uc74c \ub450 \ubb34\ud55c\uae09\uc218\ub97c \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[\\begin{align}<br \/>\nS &#038;= 1 &#8211; \\frac{1}{2} + \\frac{1}{3} &#8211; \\frac{1}{4} + \\frac{1}{5} &#8211; \\frac{1}{6} + &#8211; \\cdots , \\\\[5pt]<br \/>\nT &#038;= 1 &#8211; \\frac{1}{2} &#8211; \\frac{1}{4} + \\frac{1}{3} &#8211; \\frac{1}{6} &#8211; \\frac{1}{8} + \\frac{1}{5} &#8211; \\frac{1}{10} &#8211; \\frac{1}{12} + &#8211; &#8211; \\cdots .<br \/>\n\\end{align}\\]<br \/>\n\\(T\\)\ub294 \\(S\\)\uc758 \ud56d\uc744 \uc7ac\ubc30\uc5f4\ud55c \ubb34\ud55c\uae09\uc218\uc774\ub2e4. \ub450 \ubb34\ud55c\uae09\uc218\uc758 \ubd80\ubd84\ud569\uc744 \uac01\uac01 \\(S_n ,\\) \\(T_n \\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[\\frac{1}{2} S_{2n} =T_{3n}\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(\\left\\{ S_{2n} \\right\\}\\)\uc774 \\(S\\)\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c<br \/>\n\\[\\lim_{n\\rightarrow\\infty} T_{3n} = \\frac{1}{2}S\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc774 \ub4f1\uc2dd\uc73c\ub85c\ubd80\ud130 \\(S\\)\uac00 \uc808\ub300\uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4\ub294 \uc0ac\uc2e4\uc744 \ucd94\ub860\ud560 \uc218 \uc788\ub2e4.\n<\/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. \ub9cc\uc57d \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} \\left\\lvert a_n \\right\\rvert\\)\uc774 \uc218\ub834\ud558\uba74 \u201c\ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud55c\ub2e4(converge absolutely)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ub9cc\uc57d \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud558\uc9c0\ub9cc \uc808\ub300\uc218\ub834\ud558\uc9c0\ub294 \uc54a\uc73c\uba74 \u201c\ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc870\uac74\uc218\ub834\ud55c\ub2e4(converge conditionally)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc808\ub300\uc218\ub834\uacfc \uc870\uac74\uc218\ub834\uc758 \uad00\uacc4 \\(\\left\\{ a_n \\right\\}\\)\uc774 \uc2e4\uc218\uc5f4\uc77c \ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4. \\(\\begin{align} a_n^+ &#038;= \\begin{cases} a_n&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":"","_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":15,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6686\/revisions"}],"predecessor-version":[{"id":8061,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6686\/revisions\/8061"}],"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}]}}