{"id":6682,"date":"2021-07-20T23:54:02","date_gmt":"2021-07-20T14:54:02","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6682"},"modified":"2022-03-06T19:50:32","modified_gmt":"2022-03-06T10:50:32","slug":"infinite-series","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/infinite-series\/","title":{"rendered":"\ubb34\ud55c\uae09\uc218\uc758 \ub73b"},"content":{"rendered":"<div style=\"display: none; visibility: hidden;\">\n<style type=\"text\/css\">\n\timg.mfp-img { background-color: white; }\n<\/style>\n<p><!--\n\\[\n\\newcommand{\\vecf}{{\\mathbf{f}}}\n\\newcommand{\\vecL}{{\\mathbf{L}}}\n\\newcommand{\\vecR}{{\\mathbb{R}}}\n\\newcommand{\\imI}{\\boldsymbol{i}}\n\\]\n-->\n<\/div>\n<div class=\"box itc_intro\">\n<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 2\uc7a5 1\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><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\ubb34\ud55c\uae09\uc218\uc758 \ub73b<\/h2>\n<p>\\(\\left\\{ a_n\\right\\}\\)\uc774 \ubb34\ud55c\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \\(\\left\\{ a_n\\right\\}\\)\uc758 \ud56d\uc744 \uc21c\uc11c\ub300\ub85c \ub367\uc148\uae30\ud638\ub85c \uc5f0\uacb0\ud558\uc5ec \ub098\ud0c0\ub0b8 \uc2dd<br \/>\n\\[a_1 + a_2 + a_3 + \\cdots\\]<br \/>\n\uc744 \\(\\left\\{a_n\\right\\}\\)\uc758 <span class=\"defined\">\ubb34\ud55c\uae09\uc218<\/span>(infinite series), \ub610\ub294 \uac04\ub2e8\ud788 <span class=\"defined\">\uae09\uc218<\/span>(series)\ub77c\uace0 \ubd80\ub974\uace0, \uae30\ud638\ub85c<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n\\]<br \/>\n\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \uc774\uc640 \uac19\uc740 \uc815\uc758\uc5d0\uc11c \ubb34\ud55c\uae09\uc218\uc758 \uc815\uc758\ub294 \ud615\uc2dd\uc801(formal)\uc774\ub2e4. \uc989 \uc774 \uc815\uc758\uc5d0\uc11c \ubb34\ud55c\uae09\uc218\ub294 \uc2dd\uc758 \uac12\uc774 \uc544\ub2c8\ub77c \uc2dd \uadf8 \uc790\uccb4\ub97c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uacfc \uac19\uc740 \ud569\uc744 \uc0dd\uac01\ud558\uc790.<br \/>\n\\[S_n = \\sum_{k=1}^n a_k = a_1 + a_2 + a_3 + \\cdots + a_n .\\]<br \/>\n\uadf8\ub7ec\uba74 \uc784\uc758\uc758 \uc591\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(S_n\\)\uc774 \uc815\uc758\ub418\ubbc0\ub85c \\(\\left\\{S_n \\right\\}\\)\uc740 \uc218\uc5f4\uc774\ub2e4. \ub9cc\uc57d \\(\\left\\{S_n\\right\\}\\)\uc774 \uc218\ub834\ud558\uba74, \uc989 \uc801\ub2f9\ud55c \uc218 \\(S\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\lim_{n\\rightarrow\\infty}\\sum_{k=1}^{n} a_k = S\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\uba74 \\(S\\)\ub97c \\(\\left\\{a_n\\right\\}\\)\uc758 <span class=\"defined\">\ubb34\ud55c\uae09\uc218\uc758 \ud569<\/span>(sum)\uc774\ub77c\uace0 \ubd80\ub974\uace0, \uc774\uac83\uc744 \uae30\ud638\ub85c<br \/>\n\\[\\sum_{n=1}^{\\infty} = S\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \uc774\uc640 \uac19\uc740 \ub9e5\ub77d\uc5d0\uc11c \\(\\left\\{ S_n \\right\\}\\)\uc744 \\(\\left\\{a_n\\right\\}\\)\uc758 \ubb34\ud55c\uae09\uc218\uc758 <span class=\"defined\">\ubd80\ubd84\ud569<\/span>(partial sum)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(S\\)\ub97c \\(\\left\\{a_n\\right\\}\\)\uc758 <span class=\"defined\">\ubb34\ud55c\uae09\uc218\uc758 \uac12<\/span>(value of series)\uc774\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<p>\uc774\ub85c\uc368 \uc2dd<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n\\]<br \/>\n\uc740 \uc11c\ub85c \ub2e4\ub978 \ub450 \uac00\uc9c0\ub97c \ub3d9\uc2dc\uc5d0 \ub098\ud0c0\ub0b8\ub2e4. \ud558\ub098\ub294 \\(\\left\\{ a_n\\right\\}\\)\uc758 \ud56d\uc744 \uc21c\uc11c\ub300\ub85c \ub367\uc148\uae30\ud638\ub85c \uc5f0\uacb0\ud558\uc5ec \ub098\ud0c0\ub0b8 \ud615\uc2dd\uc801 \uc2dd\uc744 \ub098\ud0c0\ub0b4\uace0, \ub2e4\ub978 \ud558\ub098\ub294 \\(\\left\\{ a_n\\right\\}\\)\uc758 \ubb34\ud55c\uae09\uc218\uc758 \ud569\uc744 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\ub9cc\uc57d \\(n\\rightarrow\\infty\\)\uc77c \ub54c \ubd80\ubd84\ud569 \\( S_n \\)\uc774 \uc591\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud558\uac70\ub098 \uc74c\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud558\uba74, \uc774\uac83\uc744 \uae30\ud638\ub85c \uac01\uac01<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n = \\infty\\]<br \/>\n\ub610\ub294<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n = -\\infty\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 2.1.1.<\/span><br \/>\n\ub2e4\uc74c \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \uc870\uc0ac\ud558\uace0, \uc218\ub834\ud558\ub294 \uacbd\uc6b0 \uadf8 \ud569\uc744 \uad6c\ud558\uc2dc\uc624.<br \/>\n\\[\\sum_{n=1}^{\\infty} \\frac{1}{(n+1)(n+2)}\\]<br \/>\n<span class=\"proof\">\ud480\uc774.<\/span> \\(n\\ge 2\\)\uc77c \ub54c<br \/>\n\\[\\begin{align}<br \/>\nS_n &#038;= \\sum_{k=1}^n \\frac{1}{(k+1)(k+2)} \\\\[5pt]<br \/>\n&#038;= \\left( \\frac{1}{2} &#8211; \\frac{1}{3}\\right) + \\left(\\frac{1}{3} &#8211; \\frac{1}{4} \\right) + \\left(\\frac{1}{4} &#8211; \\frac{1}{5}\\right) + \\cdots + \\left( \\frac{1}{n+1} &#8211; \\frac{1}{n+2} \\right) \\\\[5pt]<br \/>\n&#038;= \\frac{1}{2} &#8211; \\frac{1}{n+2}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[\\lim_{n\\rightarrow\\infty} S_n = \\lim_{n\\rightarrow\\infty}\\left( \\frac{1}{2} &#8211; \\frac{1}{n+2} \\right) = \\frac{1}{2}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ubb38\uc81c\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\ub294 \uc218\ub834\ud558\uba70, \uadf8 \ud569\uc740 \\(\\frac{1}{2}\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 2.1.2.<\/span><br \/>\n\ub2e4\uc74c \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \uc870\uc0ac\ud558\uace0, \uc218\ub834\ud558\ub294 \uacbd\uc6b0 \uadf8 \ud569\uc744 \uad6c\ud558\uc2dc\uc624.<br \/>\n\\[\\sum_{n=1}^{\\infty} \\frac{1}{\\sqrt{n+1} + \\sqrt{n}}\\]<br \/>\n<span class=\"proof\">\ud480\uc774.<\/span> \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\uc758 \uc77c\ubc18\ud56d\uc744 \ubcc0\ud615\ud558\uba74<br \/>\n\\[\\frac{1}{\\sqrt{n+1} + \\sqrt{n}} = \\frac{\\sqrt{n+1} &#8211; \\sqrt{n}}{ \\left(\\sqrt{n+1} + \\sqrt{n}\\right)\\left(\\sqrt{n+1} &#8211; \\sqrt{n}\\right) } = \\sqrt{n+1} &#8211; \\sqrt{n}.\\]<br \/>\n\uc774 \uc2dd\uc744 \uc774\uc6a9\ud558\uc5ec \ubd80\ubd84\ud569\uc744 \uad6c\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nS_n &#038;= \\sum_{k=1}^{n} \\frac{1}{\\sqrt{k+1}+\\sqrt{k}} \\\\[5pt]<br \/>\n&#038;= \\sum_{k=1}^{n} \\left( \\sqrt{k+1} &#8211; \\sqrt{k} \\right) \\\\[5pt]<br \/>\n&#038;= \\left( \\sqrt{2} &#8211; \\sqrt{1} \\right) + \\left( \\sqrt{3} &#8211; \\sqrt{2} \\right) + \\left( \\sqrt{4} &#8211; \\sqrt{3} \\right) + \\cdots + \\left(\\sqrt{n+1} &#8211; \\sqrt{n}\\right)\\\\[5pt]<br \/>\n&#038;= \\sqrt{n+1} &#8211; 1.<br \/>\n\\end{align}\\]<br \/>\n\ub530\ub77c\uc11c<br \/>\n\\[\\lim_{n\\rightarrow\\infty} S_n = \\lim_{n\\rightarrow\\infty} \\left(\\sqrt{n+1} &#8211; 1\\right) = \\infty\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ubb38\uc81c\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\ub294 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4.\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\ubb34\ud55c\uae09\uc218\uc758 \uacc4\uc0b0<\/h2>\n<p>\ubb34\ud55c\uae09\uc218\uc758 \ubd80\ubd84\ud569\uc740 \uc218\uc5f4\uc774\uba70 \ubb34\ud55c\uae09\uc218\ub294 \ubd80\ubd84\ud569\uc758 \uadf9\ud55c\uc774\ubbc0\ub85c, \uc218\uc5f4\uc758 \uadf9\ud55c\uc758 \uc131\uc9c8\ub85c\ubd80\ud130 \ubb34\ud55c\uae09\uc218\uc758 \uc131\uc9c8\uc744 \uc5bb\uc744 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 2.1.1.<\/span><br \/>\n\ub450 \ubb34\ud55c\uae09\uc218<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n ,\\quad \\sum_{n=1}^{\\infty} b_n\\]<br \/>\n\uc774 \ubaa8\ub450 \uc218\ub834\ud558\uace0, \uadf8 \ud569\uc774 \uac01\uac01 \\(S,\\) \\(T\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\\(\\sum_{n=1}^{\\infty} \\left( ka_n \\right) = kS .\\) &nbsp;(\ub2e8, \\(k\\)\ub294 \uc0c1\uc218.)<\/li>\n<li>\\(\\sum_{n=1}^{\\infty} \\left( a_n + b_n \\right) = S+T.\\)<\/li>\n<li>\\(\\sum_{n=1}^{\\infty} \\left( a_n &#8211; b_n \\right) = S-T.\\)<\/li>\n<\/ol>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 2.1.3.<\/span><br \/>\n\\(\\sum_{n=1}^{\\infty} a_n = 3 ,\\) \\(\\sum_{n=1}^{\\infty} b_n = -4\\)\uc77c \ub54c \ub2e4\uc74c \ubb34\ud55c\uae09\uc218\uc758 \ud569\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\sum_{n=1}^{\\infty} \\left(3a_n &#8211; b_n \\right)\\)<\/li>\n<li>\\(\\sum_{n=1}^{\\infty} \\left(2a_n + 5b_n \\right)\\)<\/li>\n<\/ol>\n<p><span class=\"proof\">\ud480\uc774.<\/span><\/p>\n<ol class=\"parenthesis\">\n<li>\\[\\begin{align}\\sum_{n=1}^{\\infty} \\left(3a_n &#8211; b_n \\right) &#038;= 3\\times\\sum_{n=1}^{\\infty} a_n &#8211; \\sum_{n=1}^{\\infty}b_n \\\\[5pt] &#038;= 3\\times 3 &#8211; (-4) = 13. \\end{align}\\]<\/li>\n<li>\\[\\begin{align}\\sum_{n=1}^{\\infty} \\left(2a_n + 5b_n \\right) &#038;= 2\\times\\sum_{n=1}^{\\infty} a_n + 5\\times\\sum_{n=1}^{\\infty} b_n \\\\[5pt] &#038;= 2 \\times 3 + 5 \\times (-4) = -14.  \\end{align}\\]<\/li>\n<\/ol>\n<\/div>\n<p>\ubb34\ud55c\uae09\uc218<br \/>\n\\[\\sum_{n=1}^{\\infty} a_n\\]<br \/>\n\uc774 \uc218\ub834\ud558\uace0, \uadf8 \ud569\uc774 \\(S\\)\ub77c\uace0 \ud558\uc790. \uc77c\ubc18\ud56d \\(\\left\\{a_n \\right\\}\\)\uc758 \uadf9\ud55c\uc744 \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\lim_{n\\rightarrow\\infty} a_n<br \/>\n&#038;= \\lim_{n\\rightarrow\\infty}\\left( \\sum_{k=1}^{n} a_k &#8211; \\sum_{k=1}^{n-1} a_k \\right) \\\\[5pt]<br \/>\n&#038;= \\lim_{n\\rightarrow\\infty}\\sum_{k=1}^n &#8211; \\lim_{n\\rightarrow\\infty}\\sum_{k=1}^{n-1}a_k<br \/>\n= S-S =0.\\end{align}\\]<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 2.1.2. (\uc77c\ubc18\ud56d \ud310\uc815\ubc95)<\/span><\/p>\n<p>\ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud558\uba74 \uc77c\ubc18\ud56d \\(\\left\\{ a_n \\right\\}\\)\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2.1.4.<\/span><br \/>\n\ubb34\ud55c\uae09\uc218<br \/>\n\\[\\sum_{n=1}^{\\infty} (-1)^n\\]<br \/>\n\uc740 \ubc1c\uc0b0\ud55c\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(n\\rightarrow\\infty\\)\uc77c \ub54c \\((-1)^n\\)\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\uae30 \ub54c\ubb38\uc774\ub2e4.\n<\/p>\n<\/div>\n<p><span class=\"theorem\">\ucc38\uace0.<\/span><br \/>\n\uc815\ub9ac 2.1.2\uc758 \uc5ed\uc740 \ucc38\uc774 \uc544\ub2c8\ub2e4. \uc608\ucee8\ub300 \\(n\\rightarrow\\infty\\)\uc77c \ub54c<br \/>\n\\[\\frac{1}{\\sqrt{n+1} + \\sqrt{n}} \\,\\rightarrow\\,0\\]<br \/>\n\uc774\uc9c0\ub9cc \ubb34\ud55c\uae09\uc218<br \/>\n\\[\\sum_{n=1}^{\\infty} \\frac{1}{\\sqrt{n+1} + \\sqrt{n}}\\]<br \/>\n\uc740 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4.<br \/>\n<span class=\"qee\"><\/span><\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\ubb34\ud55c\ub4f1\ube44\uae09\uc218<\/h2>\n<p>\ub4f1\ube44\uc218\uc5f4\uc758 \ubb34\ud55c\uae09\uc218\ub97c <span class=\"defined\">\ubb34\ud55c\ub4f1\ube44\uae09\uc218<\/span> \ub610\ub294 \uac04\ub2e8\ud788 <span class=\"defined\">\ub4f1\ube44\uae09\uc218<\/span> \ub610\ub294 <span class=\"defined\">\uae30\ud558\uae09\uc218<\/span>(geometric series)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc774<br \/>\n\\[a_n = ar^{n-1} ,\\,\\, a\\ne 0\\]<br \/>\n\uc73c\ub85c \uc8fc\uc5b4\uc9c4 \ub4f1\ube44\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \\(\\left\\{ a_n \\right\\}\\)\uc758 \uae09\uc218\uc758 \ubd80\ubd84\ud569\uc744 \uad6c\ud558\uba74 \\(r\\ne 1\\)\uc77c \ub54c<br \/>\n\\[S_n = \\sum_{k=1}^n ar^{k-1} = \\frac{a(1-r^n )}{1-r}\\]<br \/>\n\uc774\uace0, \\(r = 1\\)\uc77c \ub54c<br \/>\n\\[S_n = \\sum_{k=1}^n ar^{k-1} = na\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ubd80\ubd84\ud569\uc758 \uadf9\ud55c<br \/>\n\\[\\lim_{n\\rightarrow\\infty} S_n\\]<br \/>\n\uc774 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\lvert r \\rvert < 1\\)\uc774\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 2.1.3. (\ubb34\ud55c\ub4f1\ube44\uae09\uc218\uc758 \ud569)<\/span><\/p>\n<p>\\(a\\ne 0\\)\uc77c \ub54c, \ubb34\ud55c\ub4f1\ube44\uae09\uc218<br \/>\n\\[\\sum_{n=1}^{\\infty} ar^{n-1}\\]<br \/>\n\uc774 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\lvert r \\rvert < 1\\)\uc778 \uac83\uc774\ub2e4. \uc774 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud560 \ub54c \uadf8 \ud569\uc740\n\\[\\frac{a}{1-r}\\]\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 2.1.5.<\/span><br \/>\n\ub2e4\uc74c \ubb34\ud55c\uae09\uc218\uc758 \ud569\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(1 + \\frac{2}{3} + \\left( \\frac{2}{3} \\right)^2 + \\left( \\frac{2}{3} \\right)^3 + \\cdots \\)<\/li>\n<li>\\(1 &#8211; \\frac{4}{3} + \\left( \\frac{4}{3}\\right)^2 &#8211; \\left(\\frac{4}{3}\\right)^3 + &#8211; \\cdots \\)<\/li>\n<\/ol>\n<p><span class=\"proof\">\ud480\uc774.<\/span><\/p>\n<ol class=\"parenthesis\">\n<li>\ubb38\uc81c\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\ub294 \uc77c\ubc18\ud56d\uc758 \uacf5\ube44\uac00 \\(\\frac{2}{3}\\)\uc778 \ubb34\ud55c\ub4f1\ube44\uae09\uc218\uc774\ub2e4. \uacf5\ube44\uc758 \uc808\ub313\uac12\uc774 \\(1\\)\ubcf4\ub2e4 \uc791\uc73c\ubbc0\ub85c \uc774 \ubb34\ud55c\uae09\uc218\ub294 \uc218\ub834\ud558\uba70 \uadf8 \ud569\uc740<br \/>\n\\[\\frac{1}{1-\\frac{2}{3}} = 3\\]<br \/>\n\uc774\ub2e4.<\/li>\n<li>\ubb38\uc81c\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\ub294 \uc77c\ubc18\ud56d\uc758 \uacf5\ube44\uac00 \\(-\\frac{4}{3}\\)\uc778 \ubb34\ud55c\ub4f1\ube44\uae09\uc218\uc774\ub2e4. \uacf5\ube44\uc758 \uc808\ub313\uac12\uc774 \\(1\\)\ubcf4\ub2e4 \ud06c\ubbc0\ub85c \uc774 \ubb34\ud55c\uae09\uc218\ub294 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 2.1.6.<\/span><br \/>\n\ub2e4\uc74c \ubb34\ud55c\uae09\uc218\uc758 \ud569\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac{3^n + 4^n}{5^n}\\)<\/li>\n<li>\\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac{(-2)^n + 5^n}{(-6)^n}\\)<\/li>\n<\/ol>\n<p><span class=\"proof\">\ud480\uc774.<\/span><\/p>\n<ol class=\"parenthesis\">\n<li>\\[\\begin{align}<br \/>\n\\sum_{n=1}^{\\infty} \\frac{3^n + 4^n}{5^n}<br \/>\n&#038;= \\sum_{n=1}^{\\infty}\\left(\\frac{3}{5}\\right)^n + \\sum_{n=1}^{\\infty}\\left(\\frac{4}{5}\\right)^n \\\\[5pt]<br \/>\n&#038;= \\frac{\\frac{3}{5}}{1-\\frac{3}{5}} + \\frac{\\frac{4}{5}}{1-\\frac{4}{5}} = \\frac{3}{2} + \\frac{4}{1} = \\frac{11}{2}.<br \/>\n\\end{align}\\]<\/li>\n<li>\\[\\begin{align}<br \/>\n\\sum_{n=1}^{\\infty} \\frac{(-2)^n + 5^n}{(-6)^n}<br \/>\n&#038;= \\sum_{n=1}^{\\infty}\\left(\\frac{-2}{-6}\\right)^n + \\sum_{n=1}^{\\infty}\\left(\\frac{5}{-6}\\right)^n \\\\[5pt]<br \/>\n&#038;= \\frac{\\frac{1}{3}}{1-\\frac{1}{3}} + \\frac{-\\frac{5}{6}}{1-\\left(-\\frac{5}{6}\\right)}<br \/>\n= \\frac{1}{2} + \\left(-\\frac{5}{11}\\right) = \\frac{1}{22}.<br \/>\n\\end{align}\\]<\/li>\n<\/ol>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 2.1.7.<\/span><br \/>\n\ubb34\ud55c\ub4f1\ube44\uae09\uc218\uc758 \ud569\uc744 \uc774\uc6a9\ud558\uc5ec \ub2e4\uc74c \uc21c\ud658\uc18c\uc218\ub97c \ubd84\uc218\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(0.0 \\dot{1} \\dot{2}\\)<\/li>\n<li>\\(1. \\dot{0} 1 \\dot{2}\\)<\/li>\n<\/ol>\n<p><span class=\"proof\">\ud480\uc774.<\/span><\/p>\n<ol class=\"parenthesis\">\n<li>\\[\\begin{align}<br \/>\n0.0 \\dot{1} \\dot{2}<br \/>\n&#038;= \\frac{12}{10^3} + \\frac{12}{10^5} + \\frac{12}{10^7} + \\cdots \\\\[5pt]<br \/>\n&#038;= \\frac{12}{10^3} \\times \\frac{1}{1-\\frac{1}{100}}<br \/>\n= \\frac{12}{1000} \\times \\frac{100}{99} = \\frac{2}{165}.<br \/>\n\\end{align}\\]<\/li>\n<li>\\[\\begin{align}<br \/>\n1. \\dot{0} 1 \\dot{2}<br \/>\n&#038;= 1+ \\frac{12}{1000} + \\frac{12}{1000^2} + \\frac{12}{1000^3} + \\cdots \\\\[5pt]<br \/>\n&#038;= 1+12\\times \\frac{\\frac{1}{1000}}{1-\\frac{1}{1000}}<br \/>\n= 1+12 \\times \\frac{1}{999} = \\frac{337}{333}.<br \/>\n\\end{align}\\]<\/li>\n<\/ol>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 2.1.8.<\/span><br \/>\n\uadf8\ub9bc\uacfc \uac19\uc774 \ud55c \ubcc0\uc758 \uae38\uc774\uac00 \\(4\\)\uc778 \uc815\uc0ac\uac01\ud615\uc744 \ud06c\uae30\uac00 \uac19\uc740 \\(4\\)\uac1c\uc758 \uc815\uc0ac\uac01\ud615\uc73c\ub85c \ucabc\uac20 \ub4a4 \uadf8 \uc911 \ud55c \uc870\uac01\uc5d0 \uc0c9\uc744 \uce60\ud55c\ub2e4. \uc0c9\uc774 \uce60\ud574\uc9c0\uc9c0 \uc54a\uc740 \\(3\\)\uac1c\uc758 \uc815\uc0ac\uac01\ud615 \uc870\uac01 \uc911 \ud558\ub098\ub97c \ud0dd\ud558\uc5ec \ud06c\uae30\uac00 \uac19\uc740 \\(4\\)\uac1c\uc758 \uc815\uc0ac\uac01\ud615\uc73c\ub85c \ucabc\uac20 \ub4a4 \uadf8 \uc911 \ud55c \uc870\uac01\uc5d0 \uc0c9\uc744 \uce60\ud55c\ub2e4. \uc774\uc640 \uac19\uc740 \uacfc\uc815\uc744 \ubb34\ud55c\ud788 \ubc18\ubcf5\ud560 \ub54c, \uc0c9\uc774 \uce60\ud574\uc9c4 \uc815\uc0ac\uac01\ud615 \uc870\uac01\uc758 \ub113\uc774\uc758 \ud569\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<div style=\"margin-bottom: 2em;\"><img decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2021\/07\/ex_02_01_08_square_01.png\" alt=\"\" width=\"172\" height=\"172\" class=\"aligncenter size-full wp-image-6956\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/07\/ex_02_01_08_square_01.png 687w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/07\/ex_02_01_08_square_01-300x300.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/07\/ex_02_01_08_square_01-150x150.png 150w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/07\/ex_02_01_08_square_01-585x585.png 585w\" sizes=\"(max-width: 172px) 100vw, 172px\" \/><\/div>\n<p><span class=\"proof\">\ud480\uc774.<\/span> \uadf8\ub9bc\uacfc \uac19\uc774 \uc0c9\uce60\ub41c \uc815\uc0ac\uac01\ud615\uc758 \ub113\uc774\ub97c \ud070 \uc815\uc0ac\uac01\ud615\ubd80\ud130 \ucc28\ub840\ub85c \\(S_1 ,\\) \\(S_2 ,\\) \\(S_3 ,\\) \\(\\cdots\\)\ub77c\uace0 \ud558\uc790.<\/p>\n<div><img decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2021\/07\/ex_02_01_08_square_02.png\" alt=\"\" width=\"153\" height=\"153\" class=\"aligncenter size-full wp-image-6957\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/07\/ex_02_01_08_square_02.png 612w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/07\/ex_02_01_08_square_02-300x300.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/07\/ex_02_01_08_square_02-150x150.png 150w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/07\/ex_02_01_08_square_02-585x585.png 585w\" sizes=\"(max-width: 153px) 100vw, 153px\" \/><\/div>\n<p>\uadf8\ub7ec\uba74 \\(\\left\\{S_n\\right\\}\\)\uc740 \ub4f1\ube44\uc218\uc5f4\uc774\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nS_1 &#038;= 16\\times\\frac{1}{4} = 4 ,\\\\[5pt]<br \/>\nS_2 &#038;= S_1 \\times \\frac{1}{4} = 1 ,\\\\[5pt]<br \/>\nS_3 &#038;= S_2 \\times \\frac{1}{4} = \\frac{1}{4}, \\\\[5pt]<br \/>\nS_4 &#038;= S_3 \\times \\frac{1}{4} = \\left(\\frac{1}{4}\\right)^2 ,\\\\[5pt]<br \/>\n&#038;\\,\\,\\vdots<br \/>\n\\end{align}\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \ubb34\ud55c\ub4f1\ube44\uae09\uc218\uc758 \ud569 \uacf5\uc2dd\uc5d0 \uc758\ud558\uc5ec, \uc0c9\uce60\ub41c \uc815\uc0ac\uac01\ud615\uc758 \ub113\uc774\uc758 \ud569\uc740<br \/>\n\\[\\sum_{n=1}^{\\infty}S_n = \\frac{4}{1-\\frac{1}{4}} = \\frac{16}{3}\\]<br \/>\n\uc774\ub2e4.<\/p>\n<\/div>\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=\"..\/limit-of-a-vector-sequence\">\ubca1\ud130\uc218\uc5f4\uc758 \uadf9\ud55c<\/a><\/li>\n<li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/series-of-nonnegative-terms\">\uc591\ud56d\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 1\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \ubb34\ud55c\uae09\uc218\uc758 \ub73b \\(\\left\\{ a_n\\right\\}\\)\uc774 \ubb34\ud55c\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \\(\\left\\{ a_n\\right\\}\\)\uc758 \ud56d\uc744 \uc21c\uc11c\ub300\ub85c \ub367\uc148\uae30\ud638\ub85c \uc5f0\uacb0\ud558\uc5ec \ub098\ud0c0\ub0b8 \uc2dd \\(a_1 + a_2 + a_3 + \\cdots\\) \uc744 \\(\\left\\{a_n\\right\\}\\)\uc758 \ubb34\ud55c\uae09\uc218(infinite series), \ub610\ub294 \uac04\ub2e8\ud788 \uae09\uc218(series)\ub77c\uace0 \ubd80\ub974\uace0, \uae30\ud638\ub85c \\(\\sum_{n=1}^{\\infty} a_n\\) \uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \uc774\uc640 \uac19\uc740 \uc815\uc758\uc5d0\uc11c \ubb34\ud55c\uae09\uc218\uc758 \uc815\uc758\ub294 \ud615\uc2dd\uc801(formal)\uc774\ub2e4. \uc989 \uc774 \uc815\uc758\uc5d0\uc11c \ubb34\ud55c\uae09\uc218\ub294 \uc2dd\uc758 \uac12\uc774 \uc544\ub2c8\ub77c \uc2dd \uadf8 \uc790\uccb4\ub97c \ub098\ud0c0\ub0b8\ub2e4. \ub2e4\uc74c\uacfc&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":201,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"class_list":["post-6682","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6682","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=6682"}],"version-history":[{"count":30,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6682\/revisions"}],"predecessor-version":[{"id":8383,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6682\/revisions\/8383"}],"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=6682"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}