{"id":3917,"date":"2019-06-22T03:16:28","date_gmt":"2019-06-21T18:16:28","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=3917"},"modified":"2020-05-14T10:34:03","modified_gmt":"2020-05-14T01:34:03","slug":"calculus-rearrangements-of-series","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-rearrangements-of-series\/","title":{"rendered":"\ubb34\ud55c\uae09\uc218\uc758 \uc7ac\ubc30\uc5f4"},"content":{"rendered":"<p>\uc720\ud55c \uac1c\uc758 \uc218\ub97c \ub354\ud560 \ub54c\uc5d0\ub294 \uad50\ud658\ubc95\uce59\uc774 \uc131\ub9bd\ud55c\ub2e4. \ubb34\ud55c\uae09\uc218\uc5d0\uc11c\ub294 \uc720\ud55c \uac1c\uc758 \ud56d\uc758 \uc21c\uc11c\ub97c \ubc14\uafb8\ub294 \uad50\ud658\ubc95\uce59\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ub098 \ubb34\ud55c\uae09\uc218\uc5d0\uc11c \ubb34\ud55c \uac1c\uc758 \ud56d\uc758 \uc21c\uc11c\ub97c \ubc14\uafc0 \ub54c\uc5d0\ub294 \ubcf8\ub798\uc758 \uae09\uc218\uc640\ub294 \ub2e4\ub978 \uac12\uc5d0 \uc218\ub834\ud560 \uc218\ub3c4 \uc788\uace0, \ubc1c\uc0b0\ud560 \uc218\ub3c4 \uc788\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubb34\ud55c\uae09\uc218\uc758 \ud56d\uc758 \uc21c\uc11c\ub97c \ubc14\uafc0 \ub54c \uc5b4\ub5a0\ud55c \uc77c\uc774 \ubc8c\uc5b4\uc9c0\ub294\uc9c0 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"box contentindex\">\n<h3 class=\"indextitle\">\ub0b4\uc6a9 \uc21c\uc11c<\/h3>\n<ul>\n<li><a href=\"#definition\">\uc7ac\ubc30\uc5f4\uc758 \ub73b<\/a><\/li>\n<li><a href=\"#sumofrearrangement\">\uc7ac\ubc30\uc5f4\ud55c \ubb34\ud55c\uae09\uc218\uc758 \uac12<\/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<li>\ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834 \ud310\uc815\ubc95 (<a href=\"\/blog\/articles\/calculus-convergence-tests-of-series\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<\/ul>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"definition\"><\/a><\/p>\n<h3>\uc7ac\ubc30\uc5f4\uc758 \ub73b<\/h3>\n<p>\uc7ac\ubc30\uc5f4\ub41c \uae09\uc218\ub780 \uc9c1\uad00\uc801\uc73c\ub85c\ub294 \uae09\uc218\uc758 \ud56d\uc758 \uc21c\uc11c\ub97c \ubc14\uafb8\uc5b4 \ub354\ud55c \uac83\uc744 \ub73b\ud55c\ub2e4. \uc815\ud655\ud55c \uc815\uc758\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 1. (\uae09\uc218\uc758 \uc7ac\ubc30\uc5f4; rearrangement)<\/span><\/p>\n<p>\ud568\uc218 \\(r\\)\uac00 \\(\\mathbb{N}\\)\uc73c\ub85c\ubd80\ud130 \\(\\mathbb{N}\\)\uc73c\ub85c\uc758 \uc77c\ub300\uc77c \ub300\uc751\uc774\uace0 \\(\\left\\{a_n \\right\\}\\)\uc774 \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(\\left\\{ a_{r(n)} \\right\\}\\)\uc744 \\(\\left\\{ a_n \\right\\}\\)\uc758 <span class=\"defined\">\uc7ac\ubc30\uc5f4\ub41c \uc218\uc5f4<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uba70, \\(\\sum_{n=1}^{\\infty} a_{r(n)}\\)\uc744 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc758 <span class=\"defined\">\uc7ac\ubc30\uc5f4\ub41c \uae09\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.\n<\/p>\n<\/div>\n<p>\ud568\uc218 \\(f\\)\uc5d0 \ub300\ud558\uc5ec \\(f^+\\)\uc640 \\(f^-\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nf^+ (x) &#038;= \\begin{cases}<br \/>\nf(x) \\quad &#038;\\text{if} \\,\\, f(x) \\ge 0 \\\\[6pt]<br \/>\n\\,\\,0 \\quad &#038;\\text{if} \\,\\, f(x) < 0 ,\n\\end{cases}\n\\\\[16pt]\n\nf^- (x) &#038;= \\begin{cases}\n\\,\\,\\,0 \\quad &#038;\\text{if} \\,\\, f(x) \\ge 0 \\\\[6pt]\n-f(x) \\quad &#038;\\text{if} \\,\\, f(x) < 0 .\n\\end{cases}\n\\end{align}\\]\n\uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc5d0 \ub300\ud574\uc11c\ub3c4 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.\n\\[\\begin{align}\na_n^+ (x) &#038;= \\begin{cases}\na_n \\quad &#038;\\text{if} \\,\\, a_n \\ge 0 \\\\[6pt]\n\\,\\,0 \\quad &#038;\\text{if} \\,\\, a_n < 0 ,\n\\end{cases}\n\\\\[16pt]\n\na_n^- (x) &#038;= \\begin{cases}\n\\,\\,\\,0 \\quad &#038;\\text{if} \\,\\, a_n \\ge 0 \\\\[6pt]\n-a_n \\quad &#038;\\text{if} \\,\\, a_n < 0 .\n\\end{cases}\n\\end{align}\\]\n\n\ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc758 \ubd80\ubd84\ud569 \\(s_n\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.\n\\[s_n = \\sum_{k=1}^{n} a_k = \\sum_{k=1}^{n} a_k^+ - \\sum_{k=1}^{n} a_k^-.\\]\n\uc774 \uc0ac\uc2e4\uc744 \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"lemma\">\n<p><span class=\"lemma\">\ubcf4\uc870\uc815\ub9ac 1.<\/span><\/p>\n<p>\ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"bracket marginbottom0\">\n<li>\\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud558\uba74 \\(\\sum_{n=1}^{\\infty} a_n^+\\)\uc640 \\(\\sum_{n=1}^{\\infty} a_n^-\\)\ub294 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4.<\/li>\n<li>\\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc870\uac74\uc218\ub834\ud558\uba74 \\(\\sum_{n=1}^{\\infty} a_n^+\\)\uc640 \\(\\sum_{n=1}^{\\infty} a_n^-\\)\ub294 \ubaa8\ub450 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>[1] \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \\(\\left\\{ a_n ^+\\right\\}\\)\uc640 \\(\\left\\{ a_n^- \\right\\}\\)\ub294 \ubaa8\ub450 \uc74c\uc774 \uc544\ub2cc \ud56d\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc218\uc5f4\uc774\uace0 \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\sum_{k=1}^{n} a_k^+ \\le \\sum_{k=1}^{n} \\left\\lvert a_k \\right\\rvert , \\\\[4pt]<br \/>\n\\sum_{k=1}^{n} a_k^- \\le \\sum_{k=1}^{n} \\left\\lvert a_k \\right\\rvert \\]<br \/>\n\uc774\ubbc0\ub85c \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(\\sum_{n=1}^{\\infty} a_n^+\\)\uc640 \\(\\sum_{n=1}^{\\infty} a_n^-\\)\ub294 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>[2] \ub9cc\uc57d \\(\\sum_{n=1}^{\\infty} a_n^+\\)\uc640 \\(\\sum_{n=1}^{\\infty} a_n^-\\)\uac00 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4\uba74<br \/>\n\\[\\sum_{k=1}^{n} \\left\\lvert a_k \\right\\rvert = \\sum_{k=1}^{n} a_k^+ + \\sum_{k=1}^{n} a_k^-\\]<br \/>\n\uc774\ubbc0\ub85c \\(\\sum_{n=1}^{\\infty} \\left\\lvert a_n \\right\\rvert\\)\ub3c4 \uc218\ub834\ud55c\ub2e4. \uc774\uac83\uc740 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc870\uac74\uc218\ub834\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<\/p>\n<p>\ub9cc\uc57d \\(\\sum_{n=1}^{\\infty} a_n^+\\)\uc640 \\(\\sum_{n=1}^{\\infty} a_n^-\\)\uc911 \ud558\ub098\ub294 \uc218\ub834\ud558\uace0 \ub2e4\ub978 \ud558\ub098\ub294 \ubc1c\uc0b0\ud55c\ub2e4\uba74<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} \\left\\lvert a_n \\right\\rvert\\)\ub3c4 \ubc1c\uc0b0\ud55c\ub2e4. \uc774\uac83\uc740 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<\/p>\n<p>\ub530\ub77c\uc11c \\(\\sum_{n=1}^{\\infty} a_n^+\\)\uc640 \\(\\sum_{n=1}^{\\infty} a_n^-\\)\ub294 \ubaa8\ub450 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"sumofrearrangement\"><\/a><\/p>\n<h3>\uc7ac\ubc30\uc5f4\ud55c \ubb34\ud55c\uae09\uc218\uc758 \uac12<\/h3>\n<p>\uc55e\uc758 \ubcf4\uc870\uc815\ub9ac 1\uc5d0\uc11c \ubb34\ud55c\uae09\uc218 \\(\\sum a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud560 \ub54c\uc640 \uc870\uac74\uc218\ub834\ud560 \ub54c \ub450 \ubb34\ud55c\uae09\uc218 \\(\\sum a_n^+\\)\uc640 \\(\\sum a_n^-\\)\uc758 \uc218\ub834 \ubc1c\uc0b0 \uc5ec\ubd80\uac00 \ub2ec\ub77c\uc9d0\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774 \uc0ac\uc2e4\uc744 \uc774\uc6a9\ud558\uba74 \\(\\sum a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud560 \ub54c\uc640 \uc870\uac74\uc218\ub834\ud560 \ub54c \\(\\sum a_n\\)\uc758 \uc7ac\ubc30\uc5f4\uae09\uc218\uac00 \uc5b4\ub5a0\ud55c \uc131\uc9c8\uc744 \uac16\ub294\uc9c0 \uc54c\uc544\ub0bc \uc218 \uc788\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 1. (\uc808\ub300\uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\uc758 \uc7ac\ubc30\uc5f4 \uae09\uc218)<\/span><\/p>\n<p>\ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud558\uba74 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc758 \uc7ac\ubc30\uc5f4\ub41c \ubb34\ud55c\uae09\uc218\ub3c4 \uc218\ub834\ud558\uba70, \uadf8 \uac12\uc740 \uc6d0\ub798\uc758 \ubb34\ud55c\uae09\uc218\uc758 \uac12\uacfc \ub3d9\uc77c\ud558\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\sum_{n=1}^{\\infty} b_n\\)\uc774 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc758 \uc7ac\ubc30\uc5f4\ub41c \uae09\uc218\ub77c\uace0 \ud558\uc790. \\(A = \\sum_{n=1}^{\\infty} a_n\\)\uc758 \ubd80\ubd84\ud569\uc744 \\(A_n = \\sum_{k=1}^{n} a_k\\)\ub77c\uace0 \ud558\uace0, \\(B = \\sum_{n=1}^{\\infty} b_n\\)\uc758 \ubd80\ubd84\ud569\uc744 \\(B_n = \\sum_{k=1}^{n} b_k\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc758 \uac12\uc744 \\(A\\)\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\uba3c\uc800 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc591\ud56d\uae09\uc218\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \uc784\uc758\uc758 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(a_k \\ge 0,\\) \\(b_k \\ge 0\\)\uc774\ubbc0\ub85c, \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(B_n \\le A\\)\uc774\ub2e4. \\(\\left\\{ B_n\\right\\}\\)\uc774 \ub2e8\uc870\uc99d\uac00\ud558\ub294 \uc218\uc5f4\uc774\ubbc0\ub85c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{ B_n \\right\\}\\)\uc740 \uc218\ub834\ud55c\ub2e4. \\(\\left\\{ B_n \\right\\}\\)\uc758 \uadf9\ud55c\uac12, \uc989 \\(\\sum_{n=1}^{\\infty} b_n\\)\uc758 \uac12\uc744 \\(B\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(A \\le B\\)\uc774\ub2e4. \ub450 \ubb34\ud55c\uae09\uc218\uc758 \uc5ed\ud560\uc744 \ubc14\uafb8\uba74, \\(\\sum_{n=1}^{\\infty} a_n \\)\uc740 \\(\\sum_{n=1}^{\\infty} b_n\\)\uc758 \uc7ac\ubc30\uc5f4\ub41c \uae09\uc218\uc774\ubbc0\ub85c, \uac19\uc740 \ub17c\ubc95\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(B_n \\le A\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(B\\le A\\)\uc774\ub2e4. \uc694\ucee8\ub300 \\(A=B\\)\uc774\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc591\ud56d\uae09\uc218\uac00 \uc544\ub2cc \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n = \\sum_{n=1}^{\\infty} a_n^+ &#8211; \\sum_{n=1}^{\\infty} a_n^-\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(\\sum_{n=1}^{\\infty} b_n ^+\\)\ub294 \\(\\sum_{n=1}^{\\infty} a_n ^+\\)\uc758 \uc7ac\ubc30\uc5f4 \uae09\uc218\uc774\uace0, \\(\\sum_{n=1}^{\\infty} b_n ^-\\)\ub294 \\(\\sum_{n=1}^{\\infty} a_n ^-\\)\uc758 \uc7ac\ubc30\uc5f4 \uae09\uc218\uc774\ub2e4. \ub354\uc6b1\uc774 \ub2e4\uc74c \ub124 \uae09\uc218\ub294 \ubaa8\ub450 \uc591\ud56d\uae09\uc218\uc774\ub2e4.<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n ^+, \\,\\,\\,\\sum_{n=1}^{\\infty} b_n ^+ ,\\,\\,\\, \\sum_{n=1}^{\\infty} a_n ^-, \\,\\,\\,\\sum_{n=1}^{\\infty} b_n ^-\\]<br \/>\n\ub530\ub77c\uc11c<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n^+ = \\sum_{n=1}^{\\infty} b_n^+\\]<br \/>\n\uadf8\ub9ac\uace0<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n^- = \\sum_{n=1}^{\\infty} b_n^-\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\sum_{n=1}^{\\infty}a_n = \\sum_{n=1}^{\\infty} a_n^+ &#8211; \\sum_{n=1}^{\\infty} a_n^-<br \/>\n= \\sum_{n=1}^{\\infty} b_n^+ &#8211; \\sum_{n=1}^{\\infty} b_n^- = \\sum_{n=1}^{\\infty}b_n\\]<br \/>\n\uc774\uba70, \\(\\sum_{n=1}^{\\infty} a_n\\)\uacfc \\(\\sum_{n=1}^{\\infty} b_n\\)\uc740 \uac19\uc740 \uac12\uc5d0 \uc218\ub834\ud55c\ub2e4. \ub354\uc6b1\uc774<br \/>\n\\[\\sum_{n=1}^{\\infty} \\left\\lvert a_n \\right\\rvert<br \/>\n= \\sum_{n=1}^{\\infty} a_n^+ + \\sum_{n=1}^{\\infty} a_n^-<br \/>\n= \\sum_{n=1}^{\\infty} b_n^+ + \\sum_{n=1}^{\\infty} b_n^-<br \/>\n= \\sum_{n=1}^{\\infty} \\left\\lvert b_n \\right\\rvert \\]<br \/>\n\uc774\ubbc0\ub85c \\(\\sum _{n=1}^{\\infty} b_n\\)\uc740 \uc808\ub300\uc218\ub834\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc870\uac74\uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\uc758 \uc7ac\ubc30\uc5f4\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \ub180\ub77c\uc6b4 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 2. (\uc870\uac74\uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\uc758 \uc7ac\ubc30\uc5f4 \uae09\uc218)<\/span><\/p>\n<p>\\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc870\uac74\uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\uc774\uace0 \\(A\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc9c4 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc758 \uc7ac\ubc30\uc5f4 \uae09\uc218 \uc911 \\(A\\)\uc5d0 \uc218\ub834\ud558\ub294 \uac83\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/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>\n\ubcf4\uc870\uc815\ub9ac 1\uc5d0 \uc758\ud558\uc5ec \\(\\sum_{n=1}^{\\infty} a_n^+ =\\infty\\)\uc774\ubbc0\ub85c \\(\\left\\{ a_n^+\\right\\}\\)\uc758 \ud56d\uc744 \uc55e\uc5d0\uc11c\ubd80\ud130 \uacc4\uc18d \ub354\ud558\uc5ec \\(A\\)\ubcf4\ub2e4 \ub354 \ucee4\uc9c0\ub3c4\ub85d \ud560 \uc218 \uc788\ub2e4. \uc989<br \/>\n\\[a_1 ^+ + a_2 ^+ + \\cdots + a_{n_1}^+ > A\\]<br \/>\n\uac00 \ub418\ub3c4\ub85d \ud558\ub294 \\(n_1\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7ec\ud55c \\(n_1\\) \uc911 \uac00\uc7a5 \uc791\uc740 \uac83\uc744 \ud0dd\ud558\uc790. \ub2e4\uc74c\uc73c\ub85c \\(\\sum_{n=1}^{\\infty} a_n^- = \\infty\\)\uc774\ubbc0\ub85c \\(\\left\\{ a_n ^- \\right\\}\\)\uc758 \ud56d\ub4e4\uc744 \uc55e\uc5d0\uc11c\ubd80\ud130 \uacc4\uc18d \ube7c\uc5b4\uc11c<br \/>\n\\[\\left( a_1 ^+ + a_2 ^+ + \\cdots + a_{n_1}^+ \\right) &#8211; \\left( a_{n_1 +1}^- + a_{n_1 +2}^- + \\cdots + a_{n_2}^-\\right) < A\\]\n\uac00 \ub418\ub3c4\ub85d \ud560 \uc218 \uc788\ub2e4. \uadf8\ub7ec\ud55c \\(n_2\\) \uc911 \uac00\uc7a5 \uc791\uc740 \uac83\uc744 \ud0dd\ud558\uc790. \ub2e4\uc2dc \\(\\left\\{ a_n ^+ \\right\\}\\)\uc758 \ub0a8\uc740 \ud56d\ub4e4\uc744 \uacc4\uc18d \ub354\ud558\uc5ec \ub354\ud55c \uac12\uc774 \\(A\\)\ubcf4\ub2e4 \ucee4\uc9c0\ub3c4\ub85d \ud558\uace0 \\(\\left\\{ a_n ^-\\right\\}\\)\uc758 \ub0a8\uc740 \ud56d\ub4e4\uc744 \uacc4\uc18d \ube7c\uc5b4\uc11c \uadf8 \uac12\uc774 \\(A\\)\ubcf4\ub2e4 \uc791\uc544\uc9c0\ub3c4\ub85d \ud558\ub294 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uc790. \uc989\n\\[\\begin{gather}\nP_k = a_{n_{k-1}+1}^+ + a_{n_{k-1}+2}^+ + \\cdots + a_{n_k}^+ ,\\\\[6pt]\nQ_{k+1} = a_{n_k +1}^- + a_{n_k +2}^- + \\cdots + a_{n_{k+1}}^- \n\\end{gather}\\]\n\ub77c\uace0 \ud558\uc790. \ub2e8, \uc5ec\uae30\uc11c \\(n_k\\)\uc640 \\(n_{k+1}\\)\uc740\n\\[\\begin{gather}\nP_1 - Q_2 + P_3 - Q_4 + - \\cdots + P_k > A ,\\\\[6pt]<br \/>\nP_1 &#8211; Q_2 + P_3 &#8211; Q_4 + &#8211; \\cdots + P_k &#8211; Q_{k+1} < A \n\\end{gather}\\]\n\uac00 \uc131\ub9bd\ud558\ub3c4\ub85d \ud558\ub294 \uac00\uc7a5 \uc791\uc740 \uc790\uc5f0\uc218\uc774\ub2e4. \uadf8\ub7f0\ub370\n\\[P_1 ,\\, -Q_2 ,\\, P_3 ,\\, -Q_4 ,\\, \\cdots\\]\n\ub294 \uc774 \uc21c\uc11c\ub300\ub85c \uad50\ub300\uc218\uc5f4\uc744 \uc774\ub8e8\uba70, \\(a_{n_k} \\,\\to\\, 0\\)\uc774\ubbc0\ub85c, \uc704 \uc218\uc5f4\uc758 \ubd80\ubd84\ud569\uacfc \\(A\\)\uc758 \ucc28\uc774\ub294 \\(0\\)\uc73c\ub85c \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\uc5ec\uae30\uc11c \uacb0\uad6d<br \/>\n\\[P_1 &#8211; Q_2 +P_3 -Q_4 + P_5 &#8211; Q_6  + &#8211; \\cdots\\]<br \/>\n\ub294 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc758 \uc7ac\ubc30\uc5f4 \uae09\uc218\uc774\ubbc0\ub85c \uc6d0\ud558\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<!-- h3>\uc870\uac74\uc218\ub834\ud558\ub294 \uae09\uc218\uc758 \uc7ac\ubc30\uc5f4<\/h3 -->\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>\uc720\ud55c \uac1c\uc758 \uc218\ub97c \ub354\ud560 \ub54c\uc5d0\ub294 \uad50\ud658\ubc95\uce59\uc774 \uc131\ub9bd\ud55c\ub2e4. \ubb34\ud55c\uae09\uc218\uc5d0\uc11c\ub294 \uc720\ud55c \uac1c\uc758 \ud56d\uc758 \uc21c\uc11c\ub97c \ubc14\uafb8\ub294 \uad50\ud658\ubc95\uce59\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ub098 \ubb34\ud55c\uae09\uc218\uc5d0\uc11c \ubb34\ud55c \uac1c\uc758 \ud56d\uc758 \uc21c\uc11c\ub97c \ubc14\uafc0 \ub54c\uc5d0\ub294 \ubcf8\ub798\uc758 \uae09\uc218\uc640\ub294 \ub2e4\ub978 \uac12\uc5d0 \uc218\ub834\ud560 \uc218\ub3c4 \uc788\uace0, \ubc1c\uc0b0\ud560 \uc218\ub3c4 \uc788\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubb34\ud55c\uae09\uc218\uc758 \ud56d\uc758 \uc21c\uc11c\ub97c \ubc14\uafc0 \ub54c \uc5b4\ub5a0\ud55c \uc77c\uc774 \ubc8c\uc5b4\uc9c0\ub294\uc9c0 \uc0b4\ud3b4\ubcf4\uc790. \ub0b4\uc6a9 \uc21c\uc11c \uc7ac\ubc30\uc5f4\uc758 \ub73b \uc7ac\ubc30\uc5f4\ud55c \ubb34\ud55c\uae09\uc218\uc758 \uac12 \ubbf8\ub9ac \uc54c\uc544\uc57c \ud560 \ub0b4\uc6a9 \uc218\uc5f4\uc758 \uadf9\ud55c (\uad00\ub828 \uae00) \ubb34\ud55c\uae09\uc218 (\uad00\ub828 \uae00) \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834&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":[],"class_list":["post-3917","post","type-post","status-publish","format-standard","hentry","category-calculus-ap"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/3917","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=3917"}],"version-history":[{"count":33,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/3917\/revisions"}],"predecessor-version":[{"id":4552,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/3917\/revisions\/4552"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=3917"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=3917"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=3917"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}