{"id":3762,"date":"2006-07-31T06:20:00","date_gmt":"2006-07-30T21:20:00","guid":{"rendered":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/?p=3762"},"modified":"2021-08-12T12:00:19","modified_gmt":"2021-08-12T03:00:19","slug":"studygroup2006summerchapter3","status":"publish","type":"post","link":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/2006\/07\/31\/studygroup2006summerchapter3\/","title":{"rendered":"StudyGroup2006SummerChapter3"},"content":{"rendered":"

= Sequences\/Series =<\/code> \ub0b4\uc6a9\uc694\uc57d: http:\/\/math.korea.ac.kr\/~ywkim\/courses\/2k6rudin\/rudin_ch3.pdf<\/a> <\/p>\n

\n

3\ubc88<\/h2>\n
\n

\uc810\ud654\uc2dd\uc73c\ub85c \uc8fc\uc5b4\uc9c4 \uc218\uc5f4\uc758 \uc218\ub834\uc774\uc9c0\uc694. \uc218\uc5f4\uc758 \uc218\ub834\uc5d0 \ub300\ud574\uc11c \uc544\ub294 \uac83\uc740 ”’\ub2e8\uc870\uc99d\uac00\ud558\ub294 \uc218\uc5f4\uc774 \uc704\ub85c bounded\uc774\uba74 \uc218\ub834\ud55c\ub2e4”’\ub294 \uc0ac\uc2e4\uacfc compactness\uc5d0\uc11c \ub098\uc624\ub294 ”’compact set \uc5d0\uc11c \ubb34\ud55c\uc9d1\ud569\uc740 limit point\ub97c \uac16\ub294\ub2e4”’ \uc989 ”’ $ \\mathbb{R}^n $ \uc758 bounded set\uc5d0\uc11c \uc7a1\uc740 \uc218\uc5f4\uc740 \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \uac16\ub294\ub2e4”’ \ub77c\ub294 \uac83 \ubfd0\uc785\ub2c8\ub2e4. <\/p>\n

\ub9c8\uc9c0\ub9c9 \uba85\uc81c\uac00 \uc5b4\ub5bb\uac8c \ub098\uc624\ub294\uac00\ub97c \uc798 \uc54c \ud544\uc694\uac00 \uc788\uc5b4\uc694. bounded set\uc758 closure\ub97c \uc7a1\uc73c\uba74 compact(=closed, bounded)\uc774\ub2c8\uae4c \uc774 \uc218\uc5f4\uc740 compact set\uc5d0\uc11c \uc7a1\uc740 \uac83\uc774\ub098 \ub2e4\ub984 \uc5c6\uace0 \uc774 \uc218\uc5f4\uc740 \uc720\ud55c\uac1c\uc758 \uc810\uc744 \ubc18\ubcf5\ud574\uc11c \uc6c0\uc9c1\uc774\uac70\ub098 \uc544\ub2c8\uba74 \ubb34\ud55c\ud55c \uc810\uc744 \uc6c0\uc9c1\uc774\ub2c8\uae4c \ub450 \uacbd\uc6b0 \ub2e4 \uc801\ub2f9\ud55c \ubd80\ubd84\uc218\uc5f4\uc744 \uc7a1\uc73c\uba74 \uc218\ub834\ud558\uc9c0\uc694. \uadf9\ud55c\uc810\uc740 \uc6d0\ub798\uc758 bounded set \uc548\uc5d0\ub294 \uc5c6\uc744\uc9c0\ub3c4 \ubaa8\ub974\uc9c0\ub9cc \uadf8\ub798\ub3c4 \uadf8 \uc218\uc5f4\uc740 $ \\mathbb{R}^n $ \uc5d0\uc11c \uc218\ub834\ud558\uc9c0\uc694. <\/p>\n

\ucc45\uc758 \uc810\ud654\uc2dd\uacfc \uc720\uc0ac\ud55c \ub2e4\uc74c \uc810\ud654\uc2dd\uc744 \ubcf4\uc9c0\uc694. <\/p>\n

$ s_{n+1}=\\sqrt{2+s_n},\\quad s_1=\\sqrt{2} $ <\/p>\n

\uc774 \uc2dd\uc744 \uc798 \ubcf4\uba74 $ s_n $ \uc774 \ub2e8\uc870\uc99d\uac00\ud558\ub9ac\ub77c\ub294 \uac83\uc744 guess\ud560 \uc218 \uc788\uc5b4\uc694. \uc2e4\uc81c\ub85c \uba87 \uac1c \ub123\uc5b4\uc11c \uacc4\uc0b0\ud574 \ubcf4\uc544\ub3c4 \ub418\uace0 \uc0ac\uc2e4\uc740 \uadf8\ub798\ud504\ub97c \uadf8\ub824 \ubcf4\uc544\ub3c4 \uc54c \uc218 \uc788\uc5b4\uc694.(\uc774\uac83\uc740 \uace0\ub4f1\ud559\uad50\uc2dd \ud480\uc774\uc608\uc694.) \uc6b0\ub9ac\ub294 \uc218\ub834\ud558\ub294 \uac83\uc744 \uc81c\ub300\ub85c \ubcf4\uc5ec\uc57c \ud558\ub2c8\uae4c \ub2e8\uc870\uc99d\uac00\ud558\ub294 \uac83\uc744 \ubcf4\uc785\uc2dc\ub2e4. <\/p>\n

$ s_{n+1}^2-s_n^2= (2+s_n)-(2+s_{n-1}) = s_n – s_{n-1} $ <\/p>\n

\uc774 \ub418\uc9c0\uc694. (\ucc98\uc74c\uc5d0\ub294 \uc81c\uacf1\ud558\uc9c0 \uc54a\uace0 \ube7c \ubcf4\uaca0\uc9c0\ub9cc \uadf8\ub7ec\uba74 \uacc4\uc0b0\ud558\uae30 \ud798\ub4e4\uc9c0\uc694.) \uc774\uc81c \uc624\ub978\ucabd\uacfc \uc67c\ucabd\uc744 \ube44\uad50\ud558\uba74 $ s_n $ \uc740 \ud55c\ubc88 \uc99d\uac00\ud558\uae30 \uc2dc\uc791\ud558\uba74 \uacc4\uc18d \uc99d\uac00\ud558\uaca0\uad6c\ub098 \ud558\ub294 \uac83\uc744 \uc54c \uc218 \uc788\uc5b4\uc694. \uc989, \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc4f0\uba74 \uc99d\uba85\ub418\uaca0\uc9c0\uc694? \uc6b0\uc120 $ s_2 > s_1 $ \uc778 \uac83\uc744 \ud655\uc778\ud558\uace0(\uc27d\uc9c0\uc694?), \uadf8 \ub2e4\uc74c\uc5d0 \uc704\uc758 \uacc4\uc0b0\uc73c\ub85c\ubd80\ud130 $ n $ \ubc88\uc9f8 \ud56d\uc774 \uadf8 \uc804\ud56d \ubcf4\ub2e4 \ud06c\uba74, $ n+1 $ \ubc88\uc9f8 \ud56d\ub3c4 \uadf8 \uc804\ubcf4\ub2e4 \ud06c\ub2e4\ub294 \uac83\uc744 \uc5bb\uc9c0\uc694.(\uc99d\uba85\ub05d) <\/p>\n

\uc774\uc81c \uc704\uc758 \uc218\uc5f4\uc774 \uc704\ub85c bounded\uc784\uc744 \ubcf4\uc774\ub294 \uac83\uc778\ub370. \uc774\uac83\ub3c4 \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc73c\ub85c $ s_n < 2 $ \uc774\uba74 \uae08\ubc29 $ s_{n+1} < 2 $ \uc784\uc744 \ubcf4\uc77c \uc218 \uc788\uc9c0\uc694? \uadf8\ub7ec\ub2c8\uae4c \uc774 \uc218\uc5f4\uc740 \uc218\ub834\ud558\uc9c0\uc694. <\/p>\n

\ubb38\uc81c\uc5d0\ub294 \uc5c6\uc9c0\ub9cc, \uadf9\ud55c\uc740? \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\ub294 \uac83\uc744 \uc544\ub2c8\uae4c \uc774 \uc218\uc5f4\uc758 \uadf9\ud55c\uc744 $ s $ \ub77c \ud558\uba74, \uc810\ud654\uc2dd\uc5d0\uc11c $ s $ \ub294 \uad00\uacc4\uc2dd $ s = \\sqrt{2+s} $ \ub97c \ub9cc\uc871\uc2dc\ud0a4\uc9c0\uc694. \uc774\uc81c \ud480\uc5b4\uc11c $ \\sqrt{2} $ \ubcf4\ub2e4 \ud070 (\ub2e8\uc870\uc99d\uac00\ud558\ub2c8\uae4c) $ s $ \uc758 \uac12\uc744 \uad6c\ud558\uc5ec \ubcf4\uba74 $ s=2 $ \ub77c\ub294 \uac83\uc744 \uc54c \uc218 \uc788\uc9c0\uc694. <\/p>\n

\ucc45\uc758 \ubb38\uc81c\ub3c4 \ub9c8\ucc2c\uac00\uc9c0\ub85c \ud558\uba74 \ub418\uc8e0. \ud558\uc9c0\ub9cc $ s $ \uc758 \uac12\uc744 \uad6c\ud558\ub294 \uac83\uc740 4\ucc28\ubc29\uc815\uc2dd\uc744 \ud480\uc5b4\uc57c \ud558\ub2c8\uae4c \uc5b4\ub835\uc9c0\uc694. <\/p>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

= Sequences\/Series = \ub0b4\uc6a9\uc694\uc57d: http:\/\/math.korea.ac.kr\/~ywkim\/courses\/2k6rudin\/rudin_ch3.pdf 3\ubc88 \uc810\ud654\uc2dd\uc73c\ub85c \uc8fc\uc5b4\uc9c4 \uc218\uc5f4\uc758 \uc218\ub834\uc774\uc9c0\uc694. \uc218\uc5f4\uc758 \uc218\ub834\uc5d0 \ub300\ud574\uc11c \uc544\ub294 \uac83\uc740 ”’\ub2e8\uc870\uc99d\uac00\ud558\ub294 \uc218\uc5f4\uc774 \uc704\ub85c bounded\uc774\uba74 \uc218\ub834\ud55c\ub2e4”’\ub294 \uc0ac\uc2e4\uacfc compactness\uc5d0\uc11c \ub098\uc624\ub294 ”’compact set \uc5d0\uc11c \ubb34\ud55c\uc9d1\ud569\uc740 limit point\ub97c \uac16\ub294\ub2e4”’ \uc989 ”’ $ \\mathbb{R}^n $ \uc758 bounded set\uc5d0\uc11c \uc7a1\uc740 \uc218\uc5f4\uc740 \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \uac16\ub294\ub2e4”’ \ub77c\ub294 \uac83 \ubfd0\uc785\ub2c8\ub2e4. \ub9c8\uc9c0\ub9c9 \uba85\uc81c\uac00 \uc5b4\ub5bb\uac8c \ub098\uc624\ub294\uac00\ub97c \uc798 \uc54c \ud544\uc694\uac00 \uc788\uc5b4\uc694. bounded … \ub354 \uc77d\uae30<\/a><\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-3762","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"distributor_meta":false,"distributor_terms":false,"distributor_media":false,"distributor_original_site_name":"\uae40\uc601\uc6b1","distributor_original_site_url":"https:\/\/mathematicians.korea.ac.kr\/ywkim","push-errors":false,"_links":{"self":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3762","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/comments?post=3762"}],"version-history":[{"count":1,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3762\/revisions"}],"predecessor-version":[{"id":3763,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3762\/revisions\/3763"}],"wp:attachment":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/media?parent=3762"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/categories?post=3762"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/tags?post=3762"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}