{"id":6684,"date":"2021-07-20T23:54:50","date_gmt":"2021-07-20T14:54:50","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6684"},"modified":"2021-08-05T18:16:48","modified_gmt":"2021-08-05T09:16:48","slug":"series-of-nonnegative-terms","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/series-of-nonnegative-terms\/","title":{"rendered":"\uc591\ud56d\uae09\uc218"},"content":{"rendered":"
\n

\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 2\uc7a5 2\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.  (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n

\uc2e4\uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc77c \ub54c, \ubb34\ud55c\uae09\uc218
\n\\[\\sum_{n=1}^{\\infty}a_n\\]
\n\uc744 \uc591\ud56d\uae09\uc218<\/span>(series of nonnegative terms)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

<\/p>\n

\uc720\uacc4 \ud310\uc815\ubc95<\/h2>\n

\uc2e4\uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0
\n\\[S_n = \\sum_{k=1}^{n} a_k\\]
\n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
\n\\[S_{n+1} = S_n + a_{n+1} \\ge S_n\\]
\n\uc774\ubbc0\ub85c, \ubd80\ubd84\ud569 \\(\\left\\{ S_n \\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud558\ub294 \uc218\uc5f4\uc774\ub2e4. \ub9cc\uc57d \\(\\left\\{ S_n \\right\\}\\)\uc774 \uc720\uacc4\uc774\uba74 \ub2e8\uc870\uc218\ub834\uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(n\\rightarrow\\infty\\)\uc77c \ub54c \\(\\left\\{ S_n \\right\\}\\)\uc774 \uc218\ub834\ud55c\ub2e4. \ub9cc\uc57d \\(\\left\\{ S_n \\right\\}\\)\uc774 \uc720\uacc4\uac00 \uc544\ub2c8\ub77c\uba74 \\(n\\rightarrow\\infty\\)\uc77c \ub54c \\(\\left\\{ S_n \\right\\}\\)\uc774 \uc591\uc758 \ubb34\ud55c\ub300\uc5d0 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

\uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

\n

\uc815\ub9ac 2.2.1. (\uc720\uacc4 \ud310\uc815\ubc95, Boundedness Test)<\/span><\/p>\n

\uc2e4\uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ubb34\ud55c\uae09\uc218
\n\\[\\sum_{n=1}^{\\infty} a_n\\]
\n\uc774 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ubd80\ubd84\ud569 \uc218\uc5f4 \\(\\left\\{ S_n \\right\\}\\)\uc774 \uc720\uacc4\uc778 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n

\uc608\uc81c 2.2.1.<\/span>
\n\ub2e4\uc74c \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834, \ubc1c\uc0b0 \uc5ec\ubd80\ub97c \ud310\uc815\ud558\uc2dc\uc624.<\/p>\n

\\(\\displaystyle (1) \\,\\sum_{n=1}^{\\infty} \\frac{1}{n^2} \\)
\n\\(\\displaystyle (2) \\,\\sum_{n=0}^{\\infty} \\frac{1}{n!} \\)
\n\\(\\displaystyle (3) \\,\\sum_{n=1}^{\\infty} \\frac{1}{n} \\)<\/p>\n

\ud480\uc774.<\/span><\/p>\n

(1) \\(n\\)\uc774 \\(2\\) \uc774\uc0c1\uc778 \uc790\uc5f0\uc218\uc77c \ub54c
\n\\[\\begin{align}
\n\\frac{1}{2^2} &+ \\frac{1}{3^2} + \\frac{1}{4^2} + \\cdots + \\frac{1}{n^2} \\\\[5pt]
\n&\\le \\frac{1}{1\\times 2} + \\frac{1}{2\\times 3} + \\frac{1}{3\\times 4} + \\cdots + \\frac{1}{(n-1)n} \\\\[5pt]
\n&= \\left( \\frac{1}{1} – \\frac{1}{2} \\right) + \\left( \\frac{1}{2} – \\frac{1}{3} \\right) + \\left( \\frac{1}{3} – \\frac{1}{4} \\right) + \\cdots + \\left( \\frac{1}{n-1} – \\frac{1}{n} \\right) \\\\[5pt]
\n&= 1-\\frac{1}{n}
\n\\end{align}\\]
\n\uc774\ubbc0\ub85c
\n\\[S_n = \\sum_{k=1}^n \\frac{1}{k^2} \\le 1+1 – \\frac{1}{n} \\le 2\\]
\n\uc774\ub2e4. \uc989 \ubd80\ubd84\ud569 \uc218\uc5f4 \\(\\left\\{ S_n \\right\\}\\)\uc774 \uc720\uacc4\uc774\ub2e4. \\(\\left\\{ a_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc774\ubbc0\ub85c, \uc720\uacc4 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \ubb38\uc81c\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud55c\ub2e4.<\/p>\n

(2) \\(n\\)\uc774 \uc790\uc5f0\uc218\uc77c \ub54c
\n\\[\\frac{1}{0!} + \\frac{1}{1!} + \\frac{1}{2!} + \\frac{1}{3!} + \\cdots + \\frac{1}{n!}
\n\\le \\frac{1}{1} + \\frac{1}{1} + \\frac{1}{2} + \\frac{1}{2^2} + \\cdots + \\frac{1}{2^{n-1}}
\n\\le 3\\]
\n\uc774\ubbc0\ub85c
\n\\[S_n = \\sum_{k=0}^n \\frac{1}{k!} \\le 3\\]
\n\uc774\ub2e4. \uc989 \ubd80\ubd84\ud569 \uc218\uc5f4 \\(\\left\\{ S_n \\right\\}\\)\uc774 \uc720\uacc4\uc774\ub2e4. \\(\\left\\{ a_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc774\ubbc0\ub85c, \uc720\uacc4 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \ubb38\uc81c\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud55c\ub2e4.<\/p>\n

(3) \\(m\\)\uc774 \uc790\uc5f0\uc218\uc774\uace0 \\(n=2^m\\)\uc77c \ub54c \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\begin{align}
\nS_{2^m}
\n&= \\frac{1}{1} + \\frac{1}{2} + \\left( \\frac{1}{3} + \\frac{1}{4} \\right) + \\left( \\frac{1}{5} + \\frac{1}{6} + \\frac{1}{7} + \\frac{1}{8} \\right) + \\cdots + \\left( \\frac{1}{2^{m-1}+1} + \\frac{1}{2^{m-1}+2} + \\cdots + \\frac{1}{2^m} \\right) \\\\[5pt]
\n&\\ge
\n\\frac{1}{1} + \\frac{1}{2} + \\left( \\frac{1}{4} + \\frac{1}{4} \\right) + \\left( \\frac{1}{8} + \\frac{1}{8} + \\frac{1}{8} + \\frac{1}{8} \\right) + \\cdots + \\left( \\frac{1}{2^m} + \\frac{1}{2^m} + \\cdots + \\frac{1}{2^m} \\right) \\\\[5pt]
\n&= \\frac{1}{1} + \\frac{1}{2} + \\frac{1}{2} + \\frac{1}{2} + \\cdots + \\frac{1}{2} \\\\[5pt]
\n&= \\frac{1}{1} + \\frac{1}{2} \\times m = 1 + \\frac{m}{2}.
\n\\end{align}\\]
\n\uc5ec\uae30\uc11c \\(m\\)\uc774 \uc784\uc758\uc758 \uc790\uc5f0\uc218\uc774\ubbc0\ub85c, \ubd80\ubd84\ud569 \\(S_n\\)\uc774 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ub2e4. \\(\\left\\{ a_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc774\ubbc0\ub85c, \uc720\uacc4 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \ubb38\uc81c\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\uac00 \ubc1c\uc0b0\ud55c\ub2e4.\n<\/p>\n<\/div>\n

<\/p>\n

\uc790\uc5f0\uc0c1\uc218<\/h2>\n

\uc608\uc81c 2.2.1\uc758 (2)\uc5d0\uc11c \uc0b4\ud3b4\ubcf8 \ubb34\ud55c\uae09\uc218
\n\\[\\sum_{n=0}^{\\infty} \\frac{1}{n!}\\]
\n\uc758 \uac12\uc774 \ubb34\ub9ac\uc218\uc784\uc774 \uc54c\ub824\uc838 \uc788\ub2e4. \uc774 \uac12\uc744 \uc790\uc5f0\uc0c1\uc218<\/span>(natural constant) \ub610\ub294 \uc624\uc77c\ub7ec \uc0c1\uc218<\/span>(Euler’s constant)\ub77c\uace0 \ubd80\ub974\uba70, \uc8fc\ub85c \\(\\boldsymbol{e}\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. [\u2018\uc624\uc77c\ub7ec \uc0c1\uc218\u2019\ub77c\ub294 \uc774\ub984\uc740 \\(\\gamma\\)\ub85c \ub098\ud0c0\ub0b4\ub294 \ub610 \ub2e4\ub978 \uc0c1\uc218\uac00 \uc788\uc73c\ubbc0\ub85c, \uc774 \ucc45\uc5d0\uc11c\ub294 \u2018\uc790\uc5f0\uc0c1\uc218\u2019\ub77c\ub294 \uc774\ub984\uc744 \uc0ac\uc6a9\ud558\uae30\ub85c \ud55c\ub2e4.] \uc624\uc77c\ub7ec \uc0c1\uc218\uc758 \uadfc\uc0bf\uac12\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[e = 2.718281828459045 \\cdots\\]
\n\uc790\uc5f0\uc0c1\uc218\ub97c \ub2e4\ub978 \ud615\ud0dc\ub85c \uc815\uc758\ud560 \uc218 \uc788\ub2e4.
\n\\[e_n = \\left( 1+ \\frac{1}{n} \\right)^n\\]
\n\uc774\ub77c\uace0 \ud558\uc790. \ubca0\ub974\ub204\uc774 \ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud558\uc5ec, \\(n > 2\\)\uc774\uace0 \\(x > -1\\)\uc77c \ub54c
\n\\[(1+x)^n > 1+nx\\]
\n\uc774\ubbc0\ub85c, \uc774 \uc2dd\uc5d0 \\(x = – 1\/n^2\\)\uc744 \ub300\uc785\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\left( 1- \\frac{1}{n^2} \\right)^n > 1- \\frac{1}{n}\\]
\n\uc88c\ubcc0\uc758 \uad04\ud638 \uc548\uc758 \uc2dd\uc744 \uc778\uc218\ubd84\ud574\ud558\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\left( 1+ \\frac{1}{n} \\right)^n \\left( 1- \\frac{1}{n} \\right)^n > 1 – \\frac{1}{n} \\]
\n\uc591\ubcc0\uc744 \\(\\left( 1- \\frac{1}{n}\\right)^n\\)\uc73c\ub85c \ub098\ub204\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\left( 1 + \\frac{1}{n} \\right)^n > \\left( 1+ \\frac{1}{n-1}\\right)^{n-1}\\]
\n\uc774 \ubd80\ub4f1\uc2dd\uc740 \\(e_n > e_{n-1}\\)\uc744 \ub73b\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\left\\{e_n\\right\\}\\)\uc740 \uc99d\uac00\uc218\uc5f4\uc774\ub2e4. \ub610\ud55c
\n\\[\\begin{align}
\n\\left( \\frac{1}{n} +1 \\right)^n
\n&= \\sum_{k=0}^n \\binom{n}{k} \\frac{1}{n^k} \\\\[5pt]
\n&= \\sum_{k=0}^n \\left( \\frac{n!}{k! (n-k)!} \\times \\frac{1}{n^k} \\right) \\\\[5pt]
\n&= \\sum_{k=0}^n \\left\\{ \\frac{1}{k!} \\times \\frac{n!}{(n-k)! n^k} \\right\\} \\\\[5pt]
\n&\\le \\sum_{k=0}^n \\frac{1}{k!} \\\\[5pt]
\n&= 1 + 1 + \\frac{1}{2!} + \\frac{1}{3!} + \\frac{1}{4!} + \\cdots + \\frac{1}{n!} \\\\[5pt]
\n&\\le \\sum_{k=0}^\\infty \\frac{1}{k!} = e
\n\\end{align}\\]
\n\uc774\ubbc0\ub85c \\(\\left\\{ e_n \\right\\}\\)\uc740 \\(e\\)\uc5d0 \uc758\ud558\uc5ec \uc704\ub85c \uc720\uacc4\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{ e_n \\right\\}\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n

\uc774\uc81c \\(\\left\\{ e_n \\right\\}\\)\uc758 \uadf9\ud55c\uc774 \\(e\\)\uc784\uc744 \ubcf4\uc774\uc790. \\(n\\)\uacfc \\(m\\)\uc774 \uc591\uc758 \uc815\uc218\uc774\uace0 \\(n\\ge m\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ud56d \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\n\\left( 1+ \\frac{1}{n} \\right)^n
\n&= 1 + 1 + \\frac{1}{2!}\\left(1-\\frac{1}{n}\\right) + \\frac{1}{3!}\\left( 1- \\frac{1}{n} \\right) \\left( 1- \\frac{2}{n} \\right) + \\cdots + \\frac{1}{n!}\\left( 1- \\frac{1}{n} \\right) \\cdots \\left( 1- \\frac{n-1}{n}\\right) \\\\[5pt]
\n&\\ge 1 + 1 + \\frac{1}{2!}\\left( 1- \\frac{1}{n}\\right) + \\cdots + \\frac{1}{m!}\\left(1-\\frac{1}{n} \\right) \\cdots \\left( 1- \\frac{m-1}{n}\\right).
\n\\end{align}\\]
\n\uc774 \ubd80\ub4f1\uc2dd\uc758 \uc591\ubcc0\uc5d0 \\(n\\rightarrow\\infty\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\lim_{n\\rightarrow\\infty}e_n \\ge 1 + 1 + \\frac{1}{2!} + \\frac{1}{3!} + \\cdots + \\frac{1}{m!}.\\]
\n\uc774 \ubd80\ub4f1\uc2dd\uc758 \uc591\ubcc0\uc5d0 \\(m\\rightarrow\\infty\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\lim_{n\\rightarrow\\infty}e_n \\ge e.\\]
\n\uadf8\ub828\ub370 \\(\\left\\{ e_n \\right\\}\\)\uc774 \\(e\\)\uc5d0 \uc758\ud558\uc5ec \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c, \ub2e4\uc74c \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.
\n\\[\\lim_{n\\rightarrow\\infty}e_n = e.\\]<\/p>\n

\n

\uc790\uc5f0\uc0c1\uc218<\/span>
\n\\[\\lim_{n\\rightarrow\\infty}\\left( 1 + \\frac{1}{n} \\right)^n = \\sum_{n=0}^{\\infty} \\frac{1}{n!} = e\\]\n<\/p>\n<\/div>\n

<\/p>\n

\ube44\uad50 \ud310\uc815\ubc95<\/h2>\n

\\(\\left\\{a_n\\right\\}\\)\uacfc \\(\\left\\{b_n\\right\\}\\)\uc774 \uc74c\uc774 \uc544\ub2cc \ud56d\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc218\uc5f4\uc774\uace0, \uc720\ud55c \uac1c\uc758 \\(n\\)\uc744 \uc81c\uc678\ud55c \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n \\le b_n\\)\uc774 \uc131\ub9bd\ud55c\ub2e4\uace0 \ud558\uc790. \uc989 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec [\\(n \\ge N\\)\uc774\uba74 \\(a_n \\le b_n\\)]\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(n \\ge N\\)\uc77c \ub54c
\n\\[\\sum_{k=N}^n a_k \\le \\sum_{k=N}^n b_k\\]
\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \ub9cc\uc57d \ubb34\ud55c\uae09\uc218 \\(\\sum_{n-1}^\\infty b_n\\)\uc774 \uc218\ub834\ud55c\ub2e4\uba74 \uc704 \ubd80\ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc774 \uc720\uacc4\uc774\ubbc0\ub85c, \\(n\\rightarrow\\infty\\)\uc77c \ub54c \ubd80\ub4f1\uc2dd\uc758 \uc88c\ubcc0\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n

\uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

\n

\uc815\ub9ac 2.2.2. (\ube44\uad50 \ud310\uc815\ubc95, Direct Comparison Test)<\/span><\/p>\n

\\(\\left\\{a_n\\right\\}\\)\uacfc \\(\\left\\{b_n\\right\\}\\)\uc774 \uc74c\uc774 \uc544\ub2cc \ud56d\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc218\uc5f4\uc774\uace0, \uc720\ud55c \uac1c\uc758 \\(n\\)\uc744 \uc81c\uc678\ud55c \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n \\le b_n\\)\uc774 \uc131\ub9bd\ud55c\ub2e4\uace0 \ud558\uc790.<\/p>\n

    \n
  1. \ub9cc\uc57d \\(\\sum_{n=1}^{\\infty} b_n\\)\uc774 \uc218\ub834\ud558\uba74 \\(\\sum_{n=1}^\\infty a_n\\)\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/li>\n
  2. \ub9cc\uc57d \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \ubc1c\uc0b0\ud558\uba74 \\(\\sum_{n=1}^\\infty b_n\\)\ub3c4 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
    \n

    \uc608\uc81c 2.2.2.<\/span>
    \n\uc608\uc81c 2.2.1-(2)\uc5d0 \uc758\ud558\uc5ec \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} \\frac{1}{n!}\\)\uc774 \uc218\ub834\ud55c\ub2e4. \\(p \\ge 2\\)\uc77c \ub54c
    \n\\[0\\le \\frac{1}{n^p} \\le \\frac{1}{n^2}\\]
    \n\uc774\ubbc0\ub85c, \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} \\frac{1}{n^p}\\)\uc774 \uc218\ub834\ud55c\ub2e4.\n<\/p>\n<\/div>\n

    \n

    \uc608\uc81c 2.2.3.<\/span>
    \n\uc608\uc81c 2.2.1-(3)\uc5d0\uc11c \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} \\frac{1}{n}\\)\uc774 \ubc1c\uc0b0\ud568\uc744 \ubc1d\ud614\ub2e4. \uadf8\ub7f0\ub370
    \n\\[\\frac{1}{n} \\le \\frac{1}{\\sqrt{n}}\\]
    \n\uc774\ubbc0\ub85c \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} \\frac{1}{\\sqrt{n}}\\)\uc774 \ubc1c\uc0b0\ud55c\ub2e4.\n<\/p>\n<\/div>\n

    <\/p>\n

    \uadf9\ud55c \ube44\uad50 \ud310\uc815\ubc95<\/h2>\n

    \uc218\uc5f4 \\(\\left\\{a_n\\right\\}\\)\uacfc \\(\\left\\{b_n\\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \uc591\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0
    \n\\[\\lim_{n\\rightarrow\\infty} \\frac{b_n}{a_n} = \\rho\\]
    \n\uc774\uba70, \\(\\rho\\)\uac00 \uc591\uc218\uc778 \uc0c1\uc218\ub77c\uace0 \ud558\uc790. \\(\\rho +1\\)\uc774 \\(\\rho\\)\ubcf4\ub2e4 \ud070 \uc2e4\uc218\uc774\ubbc0\ub85c, \uc720\ud55c \uac1c\ub97c \uc81c\uc678\ud55c \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[\\frac{b_n}{a_n} \\le \\rho +1\\]
    \n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc989 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(n\\ge N\\)\uc77c \ub54c
    \n\\[\\frac{b_n}{a_n} \\le \\rho +1\\]
    \n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc591\ubcc0\uc5d0 \\(\\rho +1\\)\uc744 \uacf1\ud558\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
    \n\\[b_n \\le (\\rho +1)a_n .\\]
    \n\ub9cc\uc57d \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4\uba74, \uc704 \ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud558\uc5ec \\(\\sum_{n=1}^{\\infty}b_n\\)\ub3c4 \uc218\ub834\ud55c\ub2e4. \ud55c\ud3b8
    \n\\[\\lim_{n\\rightarrow\\infty} \\frac{a_n}{b_n} = \\frac{1}{\\rho}\\]
    \n\ub610\ud55c \uc591\uc218\uc778 \uc0c1\uc218\uc774\ubbc0\ub85c, \ub9cc\uc57d \\(\\sum_{n=1}^{\\infty} b_n\\)\uc774 \uc218\ub834\ud55c\ub2e4\uba74 \uc55e\uc5d0\uc11c\uc640 \uac19\uc740 \ub17c\uc99d\uc5d0 \uc758\ud558\uc5ec \\(\\sum_{n=1}^{\\infty} a_n\\)\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/p>\n

    \uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 2.2.3. (\uadf9\ud55c \ube44\uad50 \ud310\uc815\ubc95, Limit Comparison Test)<\/span><\/p>\n

    \uc218\uc5f4 \\(\\left\\{a_n\\right\\}\\)\uacfc \\(\\left\\{b_n\\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \uc591\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0
    \n\\[\\lim_{n\\rightarrow\\infty} \\frac{b_n}{a_n} = \\rho\\]
    \n\uc774\uba70, \\(\\rho\\)\uac00 \uc591\uc218\uc778 \uc0c1\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} b_n\\)\uc774 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n

    \n

    \uc608\uc81c 2.2.4.<\/span>
    \n\ub2e4\uc74c \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \ud310\uc815\ud574 \ubcf4\uc790.
    \n\\[\\sum_{n=1}^{\\infty} \\frac{4n^2 – n +3}{n^3 +2n}.\\]
    \n\ubd84\ubaa8\uc758 \ucc28\uc218\uac00 \ubd84\uc790\uc758 \ucc28\uc218\ubcf4\ub2e4 \\(1\\) \ud06c\ub2e4\ub294 \uc0ac\uc2e4\uc5d0 \uc8fc\ubaa9\ud558\uc790.
    \n\\[a_n = \\frac{4n^2 – n+3}{n^3 +2n} ,\\quad b_n = \\frac{1}{n}\\]
    \n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(a_n > 0,\\) \\(b_n > 0\\)\uc774\uace0,
    \n\\[\\frac{a_n}{b_n} = \\frac{4n^3 -n^2 +3n}{n^3 +2n} \\,\\, \\rightarrow \\,\\, 4 \\quad \\text{ as } \\quad n\\rightarrow\\infty\\]
    \n\uc774\ub2e4. \uadf8\ub7f0\ub370 \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty}\\frac{1}{n}\\)\uc774 \ubc1c\uc0b0\ud558\ubbc0\ub85c \\(\\sum_{n=1}^{\\infty} \\frac{4n^2 – n +3}{n^3 +2n}\\)\ub3c4 \ubc1c\uc0b0\ud55c\ub2e4.\n<\/p>\n<\/div>\n

    <\/p>\n

    \ube44 \ud310\uc815\ubc95<\/h2>\n

    \ube44\uad50 \ud310\uc815\ubc95\uc774\ub098 \uadf9\ud55c \ube44\uad50 \ud310\uc815\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \ud310\uc815\ud560 \ub54c\ub294 \uc218\ub834 \uc5ec\ubd80\uac00 \uc54c\ub824\uc9c4 \ub2e4\ub978 \ubb34\ud55c\uae09\uc218\uac00 \ud544\uc694\ud588\ub2e4. \ud558\uc9c0\ub9cc \uc218\ub834 \uc5ec\ubd80\uac00 \uc54c\ub824\uc9c4 \ub2e4\ub978 \ubb34\ud55c\uae09\uc218\uac00 \ud544\uc694\ud558\uc9c0 \uc54a\uc740 \ud310\uc815\ubc95\uc774 \uc788\ub2e4. \ube44 \ud310\uc815\ubc95\uacfc \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc774 \uadf8\uc640 \uac19\uc740 \ud310\uc815\ubc95\uc774\ub2e4.<\/p>\n

    \uc218\uc5f4 \\(\\left\\{a_n\\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \uc591\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0
    \n\\[\\lim_{n\\rightarrow\\infty} \\frac{a_{n+1}}{a_n} = \\rho\\]
    \n\uc774\uba70 \\(\\rho\\)\uac00 \uc0c1\uc218\ub77c\uace0 \ud558\uc790. \uc774\uc81c \\(\\rho\\)\uac00 \\(1\\)\ubcf4\ub2e4 \uc791\uc740 \uacbd\uc6b0\uc640 \\(1\\)\ubcf4\ub2e4 \ud070 \uacbd\uc6b0\ub85c \ub098\ub204\uc5b4 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

    \uba3c\uc800 \\(\\rho < 1\\)\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \\(\\rho < r < 1\\)\uc778 \uc2e4\uc218 \\(r\\)\ub97c \ud0dd\ud558\uc790. \\(n\\rightarrow\\infty\\)\uc77c \ub54c \\(a_{n+1}\/a_n\\)\uc774 \\(\\rho\\)\uc5d0 \uc218\ub834\ud558\uace0 \\(r > \\rho\\)\uc774\ubbc0\ub85c, \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[\\frac{a_{n+1}}{a_n} \\le r\\]
    \n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc989 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(n\\ge N\\)\uc77c \ub54c
    \n\\[\\frac{a_{n+1}}{a_n} \\le r\\]
    \n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc591\ubcc0\uc5d0 \\(a_n\\)\uc744 \uacf1\ud558\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
    \n\\[a_{n+1} \\le ra_n .\\]
    \n\uc5ec\uae30\uc11c \\(n\\)\uc744 \\(N,\\) \\(N+1,\\) \\(N+2,\\) \\(\\cdots\\)\uc73c\ub85c \ubc14\uafb8\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
    \n\\[\\begin{align}
    \na_{N+1} &\\le ra_N ,\\\\[5pt]
    \na_{N+2} &\\le ra_{N+1} \\le r^2 a_N , \\\\[5pt]
    \na_{N+3} &\\le ra_{N+2} \\le r^3 a_N , \\\\[5pt]
    \n&\\,\\,\\vdots \\\\[5pt]
    \na_{N+k} &\\le ra_{N+k-1} \\le r^k a_N .
    \n\\end{align}\\]
    \n\uadf8\ub7f0\ub370 \\(0 < r < 1\\)\uc774\ubbc0\ub85c \\(\\sum_{k=1}^{\\infty} r^k a_N\\)\uc774 \uc218\ub834\ud55c\ub2e4. \ub610\ud55c\n\\[\\begin{align}\nS_{N+k}\n&= a_1 + a_2 + \\cdots + a_{N+k} \\\\[5pt]\n&= a_1 + a_2 + \\cdots + a_N + ( a_{N+1} + a_{N+2} + \\cdots + a_{N+k}) \\\\[5pt]\n&\\le a_1 + a_2 + \\cdots + a_N + (ra_N + r^2 a_N + \\cdots + r^k a_N ) \\\\[5pt]\n&\\le a_1 + a_2 + \\cdots + a_N + \\sum_{k=1}^\\infty r^k a_N .\n\\end{align}\\]\n\uadf8\ub7ec\ubbc0\ub85c \\(\\left\\{ S_n \\right\\}\\)\uc774 \uc720\uacc4\uc774\uba70, \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n

    \ub2e4\uc74c\uc73c\ub85c \\(\\rho > 1\\)\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \uc774\ub54c\ub294 \\(\\left\\{ a_n \\right\\}\\)\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

    \uc774\ub85c\uc368 \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 2.2.4. (\ube44 \ud310\uc815\ubc95, Ratio Test)<\/span><\/p>\n

    \uc218\uc5f4 \\(\\left\\{a_n\\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \uc591\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0
    \n\\[\\lim_{n\\rightarrow\\infty} \\frac{a_{n+1}}{a_n} = \\rho\\]
    \n\uc774\uba70 \\(\\rho\\)\uac00 \uc0c1\uc218\ub77c\uace0 \ud558\uc790.<\/p>\n

      \n
    1. \ub9cc\uc57d \\(\\rho < 1\\)\uc774\uba74 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/li>\n
    2. \ub9cc\uc57d \\(\\rho > 1\\)\uc774\uba74 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n
    3. \ub9cc\uc57d \\(\\rho = 1\\)\uc774\uba74 \uc774 \ud310\uc815\ubc95\uc73c\ub85c \\(\\sum_{n=1}^{\\infty} a_n\\)\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \ud310\uc815\ud560 \uc218 \uc5c6\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
      \n

      \uc608\uc81c 2.2.5.<\/span>
      \n\\(x > 0\\)\uc77c \ub54c \ub2e4\uc74c \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \ud310\uc815\ud574 \ubcf4\uc790.
      \n\\[\\sum_{n=1}^{\\infty} \\frac{x^n}{n!}.\\]
      \n\\(a_n = \\frac{x^n}{n!}\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
      \n\\[\\lim_{n\\rightarrow\\infty} \\frac{a_{n+1}}{a_n} = \\lim_{n\\rightarrow\\infty}\\frac{x}{n+1} = 0 < 1\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\sum_{n=1}^{\\infty} \\frac{x^n}{n!}\\)\uc774 \uc218\ub834\ud55c\ub2e4.\n<\/p>\n<\/div>\n

      \n

      \uc608\uc81c 2.2.6.<\/span>
      \n\ub2e4\uc74c \ub450 \ubb34\ud55c\uae09\uc218\ub97c \uc0b4\ud3b4\ubcf4\uc790.
      \n\\[\\sum_{n=1}^{\\infty} \\frac{1}{n} , \\quad \\sum_{n=1}^{\\infty} \\frac{1}{n^2} .\\]
      \n\uc608\uc81c 2.2.1\uc5d0\uc11c \uccab\uc9f8 \ubb34\ud55c\uae09\uc218\uac00 \ubc1c\uc0b0\ud558\uace0 \ub458\uc9f8 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud568\uc744 \ubc1d\ud614\ub2e4.<\/p>\n

      \\(a_n = \\frac{1}{n} ,\\) \\(b_n = \\frac{1}{n^2}\\)\uc774\ub77c\uace0 \ud558\uba74
      \n\\[\\lim_{n\\rightarrow\\infty} \\frac{a_{n+1}}{a_n} = 1 ,\\quad \\lim_{n\\rightarrow\\infty} \\frac{b_{n+1}}{b_n} = 1\\]
      \n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ube44 \ud310\uc815\ubc95\uc5d0\uc11c \\(\\rho=1\\)\uc77c \ub54c \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\ub294 \uc218\ub834\ud560 \uc218\ub3c4 \uc788\uace0 \ubc1c\uc0b0\ud560 \uc218\ub3c4 \uc788\ub2e4.\n<\/p>\n<\/div>\n

      <\/p>\n

      \uc81c\uacf1\uadfc \ud310\uc815\ubc95<\/h2>\n

      \uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0
      \n\\[\\lim_{n\\rightarrow\\infty} \\sqrt[n]{a_n} = \\rho\\]
      \n\uc774\uba70 \\(\\rho\\)\uac00 \uc0c1\uc218\ub77c\uace0 \ud558\uc790. \uc774\uc81c \\(\\rho\\)\uac00 \\(1\\)\ubcf4\ub2e4 \uc791\uc740 \uacbd\uc6b0\uc640 \\(1\\)\ubcf4\ub2e4 \ud070 \uacbd\uc6b0\ub85c \ub098\ub204\uc5b4 \uc0b4\ud3b4\ubcf4\uc790.<\/p\n\n\n\n

      \uba3c\uc800 \\(\\rho < 1\\)\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \\(\\rho < r < 1\\)\uc778 \uc2e4\uc218 \\(r\\)\ub97c \ud0dd\ud558\uc790. \\(n\\rightarrow\\infty\\)\uc77c \ub54c \\(\\sqrt[n]{a_n}\\)\uc774 \\(\\rho\\)\uc5d0 \uc218\ub834\ud558\uace0 \\(r < \\rho\\)\uc774\ubbc0\ub85c, \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(n\\ge N\\)\uc77c \ub54c\n\\[\\sqrt[n]{a_n} \\le r\\]\n\uc989\n\\[a_n \\le r^n\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc5ec\uae30\uc11c \\(n\\)\uc744 \\(N,\\) \\(N+1,\\) \\(N+2,\\) \\(\\cdots\\)\uc73c\ub85c \ubc14\uafb8\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.\n\\[\\begin{align}\na_N &\\le r^N ,\\\\[5pt]\na_{N+1} &\\le r^{N+1} ,\\\\[5pt]\na_{N+2} &\\le r^{N+2} ,\\\\[5pt]\n&\\,\\,\\vdots \\\\[5pt]\na_{N+k} &\\le r^{N+k}.\n\\end{align}\\]\n\uadf8\ub7f0\ub370 \\(0 < r < 1\\)\uc774\ubbc0\ub85c \\(\\sum_{k=0}^{\\infty} r^{N+k}\\)\uc774 \uc218\ub834\ud55c\ub2e4. \ub610\ud55c\n\\[\\begin{align}\nS_{N+k}\n&= a_1 + a_2 + \\cdots + a_{N-1} + (a_N + a_{N+1} + \\cdots + a_{N+k}) \\\\[5pt]\n&\\le a_1 + a_2 + \\cdots + a_{N-1} + r^N + r^{N+1} +\\cdots + r^{N+k} \\\\[5pt]\n&\\le a_1 + a_2 + \\cdots + a_{N-1} + \\sum_{k=0}^{\\infty} r^{N+k}\n\\end{align}\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\left\\{ S_n \\right\\}\\)\uc774 \uc720\uacc4\uc774\uba70, \\(\\sum_{n=1}^\\infty a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n

      \ub2e4\uc74c\uc73c\ub85c \\(\\rho > 1\\)\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \uc774\ub54c\ub294 \\(\\left\\{ a_n \\right\\}\\)\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

      \uc774\ub85c\uc368 \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

      \n

      \uc815\ub9ac 2.2.5. (\uc81c\uacf1\uadfc \ud310\uc815\ubc95, Root Test)<\/span><\/p>\n

      \uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0
      \n\\[\\lim_{n\\rightarrow\\infty} \\sqrt[n]{a_n} = \\rho\\]
      \n\uc774\uba70 \\(\\rho\\)\uac00 \uc0c1\uc218\ub77c\uace0 \ud558\uc790.<\/p>\n

        \n
      1. \ub9cc\uc57d \\(\\rho < 1\\)\uc774\uba74 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/li>\n
      2. \ub9cc\uc57d \\(\\rho > 1\\)\uc774\uba74 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n
      3. \ub9cc\uc57d \\(\\rho = 1\\)\uc774\uba74 \uc774 \ud310\uc815\ubc95\uc73c\ub85c \\(\\sum_{n=1}^{\\infty} a_n\\)\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \ud310\uc815\ud560 \uc218 \uc5c6\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
        \n

        \uc608\uc81c 2.2.7.<\/span>
        \n\\(x\\ge 0\\)\uc77c \ub54c, \ub2e4\uc74c \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \ud310\uc815\ud574 \ubcf4\uc790.
        \n\\[ 1 + 2x + x^2 + 2x^3 + x^4 + 2x^5 + \\cdots . \\]
        \n\ub9cc\uc57d \\(x=0\\)\uc774\uba74 \ubb38\uc81c\uc758 \ubb34\ud55c\uae09\uc218\uac00 \ub2f9\uc5f0\ud788 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x > 0\\)\uc778 \uacbd\uc6b0\ub97c \uc870\uc0ac\ud558\uc790.<\/p>\n

        \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\uc758 \\(n\\)\uc9f8 \ud56d\uc744 \\(a_n\\)\uc774\ub77c\uace0 \ud558\uc790. \ubd84\uc218\uc2dd
        \n\\[\\frac{a_{n+1}}{a_n}\\]
        \n\uc758 \uac12\uc774 \\(2x\\)\uc640 \\(\\frac{1}{2} x\\)\uac00 \ubc88\uac08\uc544 \ub098\ud0c0\ub098\uae30 \ub54c\ubb38\uc5d0 \ube44 \ud310\uc815\ubc95\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc5c6\ub2e4. \ub300\uc2e0 \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc744 \uc0ac\uc6a9\ud558\uc790. \\(n\\)\uc774 \ud640\uc218\uc77c \ub54c
        \n\\[\\sqrt[n]{a_n} = \\sqrt[n]{2x^n} = \\sqrt[n]{2}x\\]
        \n\uc774\uace0, \\(n\\)\uc774 \uc9dd\uc218\uc77c \ub54c
        \n\\[\\sqrt[n]{a_n} = \\sqrt[n]{x^n} = x\\]
        \n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
        \n\\[\\lim_{n\\rightarrow\\infty} \\sqrt[n]{x_n} = x\\]
        \n\uc774\ub2e4. \ub530\ub77c\uc11c \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(x < 1\\)\uc77c \ub54c \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud558\uace0 \\(x > 1\\)\uc77c \ub54c \ubb34\ud55c\uae09\uc218\uac00 \ubc1c\uc0b0\ud55c\ub2e4. \\(x=1\\)\uc77c \ub54c\ub294 \ubb34\ud55c\uae09\uc218\uc758 \ud56d\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c \ubb34\ud55c\uae09\uc218\uac00 \ubc1c\uc0b0\ud55c\ub2e4.\n<\/p>\n<\/div>\n

        \ucc38\uace0.<\/span> \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc73c\ub85c \uc218\ub834 \uc5ec\ubd80\uac00 \ud310\uc815\ub418\uc9c0 \uc54a\ub294 \ubb34\ud55c\uae09\uc218\uac00 \uc788\ub2e4. \uc608\ucee8\ub300
        \n\\[\\sum_{n=1}^{\\infty} \\frac{1}{n}\\]
        \n\uc744 \uc0b4\ud3b4\ubcf4\uc790. \\[\\lim_{n\\rightarrow\\infty} \\sqrt[n]{\\frac{1}{n}} = 1\\]
        \n\uc774\ubbc0\ub85c \uc774 \ubb34\ud55c\uae09\uc218\ub294 \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc73c\ub85c \ud310\uc815\ub418\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n

        \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} 1\\)\uc740 \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc73c\ub85c \ud310\uc815\ub418\uc9c0 \uc54a\ub294 \ub354 \uac04\ub2e8\ud55c \uc608\uc774\ub2e4.
        \n<\/span><\/p>\n

        <\/p>\n

        \ucf54\uc2dc\uc758 \uc751\uc9d1 \ud310\uc815\ubc95<\/h2>\n

        \uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\left\\{ a_n \\right\\}\\)\uc774 \ub2e8\uc870\uac10\uc18c\ud558\uba70 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uc774\uc81c \ub450 \ubb34\ud55c\uae09\uc218
        \n\\[\\sum_{n=1}^{\\infty} a_n , \\quad \\sum_{k=1}^{\\infty} 2^k a_{2^k}\\]
        \n\uc758 \uad00\uacc4\ub97c \ubc1d\ud788\uc790.<\/p>\n

        \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74
        \n\\[\\begin{align}
        \na_1 + \\frac{1}{2} \\sum_{k=1}^n 2^k a_{2^k}
        \n&= a_1 + a_2 + 2a_4 + 4a_8 + \\cdots + 2^{n-1} a_{2^n} \\\\[5pt]
        \n&\\le a_1 + a_2 + a_3 + \\cdots + a_{2^{n-1}+1} + \\cdots + a_{2^n -1} + a_{2^n} \\\\[5pt]
        \n&\\le \\sum_{k=1}^{\\infty} a_k
        \n\\end{align}\\]
        \n\uc774\ubbc0\ub85c \\(\\sum_{k=1}^{\\infty} 2^k a_{2^k}\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n

        \uc5ed\uc73c\ub85c, \\(\\sum_{k=1}^{\\infty} 2^k a_{2^k}\\)\uc774 \uc218\ub834\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74
        \n\\[\\begin{align}
        \n\\sum_{k=1}^n a_k
        \n&= a_1 + a_2 + a_3 + \\cdots + a_n \\\\[5pt]
        \n&\\le a_1 + (a_2 + a_3 ) + (a_4 + a_5 + a_6 + a_7 ) + \\cdots + (a_{2^n} + a_{2^n +1} + \\cdots + a_{2^{n+1} -1} ) \\\\[5pt]
        \n&\\le a_1 + 2a_2 + 4a_4 + \\cdots + 2^n a_{2^n} \\\\[5pt]
        \n&\\le a_1 + \\sum_{k=1}^\\infty 2^k a_{2^k}
        \n\\end{align}\\]
        \n\uc774\ubbc0\ub85c \\(\\sum_{k=1}^{\\infty} a_k\\)\uac00 \uc218\ub834\ud55c\ub2e4.<\/p>\n

        \uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

        \n

        \uc815\ub9ac 2.2.6. (\ucf54\uc2dc\uc758 \uc751\uc9d1 \ud310\uc815\ubc95, Cauchy’s Condensation Test)<\/span><\/p>\n

        \uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc774 \ub2e8\uc870\uac10\uc18c\ud558\uba70 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
        \n\\[\\sum_{n=1}^{\\infty} a_n \\]
        \n\uc774 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740
        \n\\[\\sum_{n=1}^{\\infty} 2^n a_{2^n} \\]
        \n\uc774 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n

        \n

        \uc608\uc81c 2.2.8.<\/span>
        \n\ub2e4\uc74c \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \ud310\uc815\ud558\uc2dc\uc624.<\/p>\n

        \\(\\displaystyle (1) \\, \\sum_{n=1}^\\infty \\frac{1}{n}\\)
        \n\\(\\displaystyle (2) \\, \\sum_{n=2}^\\infty \\frac{1}{n\\ln n}\\)
        \n\\(\\displaystyle (3) \\, \\sum_{n=2}^\\infty \\frac{1}{n(\\ln n)^2}\\)
        \n\\(\\displaystyle (4) \\, \\sum_{n=9}^\\infty \\frac{1}{n\\ln(\\ln n)}\\)\n<\/p>\n

        \ud480\uc774.<\/span><\/p>\n

        (1) \uc218\uc5f4 \\(\\left\\{ \\frac{1}{n} \\right\\}\\)\uc740 \uac10\uc18c\ud558\uba70 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7f0\ub370
        \n\\[\\sum_{n=1}^{\\infty} \\left( 2^n \\cdot \\frac{1}{2^n} \\right) = \\sum_{n=1}^{\\infty} 1 = \\infty\\]
        \n\uc774\ubbc0\ub85c, \uc751\uc9d1 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(\\sum_{n=1}^{\\infty} \\frac{1}{n}\\)\uc774 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

        (2) \uc218\uc5f4 \\(\\left\\{ \\frac{1}{n\\ln n} \\right\\}\\)\uc740 \uac10\uc18c\ud558\uba70 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7f0\ub370 (1)\uc5d0 \uc758\ud558\uc5ec
        \n\\[ \\sum_{n=2}^\\infty \\left( 2^n \\cdot \\frac{1}{2^n \\ln 2^n} \\right)
        \n= \\sum_{n=2}^{\\infty} \\frac{1}{ n \\ln 2} = \\infty \\]
        \n\uc774\ubbc0\ub85c, \uc751\uc9d1 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(\\sum_{n=2}^{\\infty} \\frac{1}{\\ln n}\\)\uc774 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

        (3) \uc218\uc5f4 \\(\\left\\{ \\frac{1}{n (\\ln n)^2} \\right\\}\\)\uc740 \uac10\uc18c\ud558\uba70 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7f0\ub370
        \n\\[ \\sum_{n=2}^{\\infty} \\left( 2^n \\cdot \\frac{1}{2^n (\\ln 2^n )^2} \\right) = \\sum_{n=2}^{\\infty} \\frac{1}{n^2 (\\ln 2)^2} = \\frac{1}{(\\ln 2)^2} \\sum_{n=2}^{\\infty} \\frac{1}{n^2} \\]
        \n\uc774 \uc218\ub834\ud558\ubbc0\ub85c, \uc751\uc9d1 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(\\sum_{n=2}^{\\infty} \\frac{1}{n (\\ln n)^2}\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n

        (4) \uc218\uc5f4 \\(\\left\\{ \\frac{1}{n\\ln (\\ln n)} \\right\\}\\)\uc740 \uac10\uc18c\ud558\uba70 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7f0\ub370
        \n\\[ \\sum_{n=9}^{\\infty} \\left( 2^n \\frac{1}{2^n \\ln ( \\ln 2^n )}\\right)
        \n= \\sum_{n=9}^{\\infty} \\frac{1}{\\ln (n\\ln 2)} =
        \n\\sum_{n=9}^{\\infty} \\frac{1}{\\ln n + \\ln (\\ln 2)} =\\infty \\]
        \n\uc774\ubbc0\ub85c, \uc751\uc9d1 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(\\sum_{n=9}^{\\infty} \\frac{1}{n \\ln (\\ln n)}\\)\uc774 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc608\uc81c 2.2.9.<\/span>
        \n\ubb34\ud55c\uae09\uc218
        \n\\[\\sum_{n=1}^\\infty \\frac{1}{n^p}\\]
        \n\ub97c \\(\\boldsymbol{p}\\)-\uae09\uc218<\/span>(p-series)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc774 \\(p > 1\\)\uc778 \uac83\uc784\uc744 \ubc1d\ud788\uc2dc\uc624.<\/p>\n

        \ud480\uc774.<\/span> \ub9cc\uc57d \\(p \\le 0\\)\uc774\uba74 \\(\\left\\{ \\frac{1}{n^p}\\right\\}\\)\uac00 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\sum_{n=1}^\\infty \\frac{1}{n^p}\\)\uc774 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

        \uc774\uc81c \\(p > 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\left\\{ \\frac{1}{n^p}\\right\\}\\)\uc740 \uac10\uc18c\ud558\uace0 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \ucf54\uc2dc \uc751\uc9d1\ud310\uc815\ubc95\uc744 \uc0ac\uc6a9\ud558\uae30 \uc704\ud558\uc5ec \ub2e4\uc74c \uc2dd\uc744 \uc0b4\ud3b4\ubcf4\uc790.
        \n\\[\\sum_{n=1}^\\infty \\left( 2^n \\cdot \\frac{1}{(2^n )^p} \\right) = \\sum_{n=1}^\\infty 2^{n(1-p)} = \\sum_{n=1}^{\\infty} (2^{1-p})^n\\]
        \n\ub9c8\uc9c0\ub9c9 \ubb34\ud55c\uae09\uc218\ub294 \ubb34\ud55c\ub4f1\ube44\uae09\uc218\uc774\uba70 \uacf5\ube44\uac00 \\(2^{1-p}\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc774 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(2^{1-p} < 1\\)\uc774\ub2e4. \uc774 \ubd80\ub4f1\uc2dd\uc744 \ud480\uba74 \\(1-p < 0,\\) \uc989 \\(p > 1\\)\uc744 \uc5bb\ub294\ub2e4.\n<\/p>\n<\/div>\n

        <\/p>\n

        \uc81c\uacf1\uadfc \uc0c1\uadf9\ud55c \ud310\uc815\ubc95<\/h2>\n

        \uc0c1\uadf9\ud55c\uc744 \uc774\uc6a9\ud558\uc5ec \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc744 \ub354 \ub2e4\uc591\ud55c \ubb34\ud55c\uae09\uc218\uc758 \ud310\uc815\uc5d0 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub3c4\ub85c \uac1c\uc120\ud560 \uc218 \uc788\ub2e4.<\/p>\n

        \uc218\uc5f4 \\(\\left\\{a_n\\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0
        \n\\[\\varlimsup_{n\\rightarrow\\infty} \\sqrt[n]{a_n} = \\rho\\]
        \n\uc774\uba70 \\(\\rho\\)\uac00 \uc0c1\uc218\ub77c\uace0 \ud558\uc790.<\/p>\n

        \\(\\rho < 1\\)\uc778 \uacbd\uc6b0 \\(\\rho < r < 1\\)\uc778 \uc2e4\uc218 \\(r\\)\ub97c \ud0dd\ud558\uc790. \\(\\left\\{ \\sqrt[n]{a_n}\\right\\}\\)\uc758 \ubd80\ubd84\uc218\uc5f4 \uc911\uc5d0\uc11c \\(\\rho\\)\ubcf4\ub2e4 \ub354 \ud070 \uac12\uc5d0 \uc218\ub834\ud558\ub294 \uac83\uc740 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c, \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\\sqrt[n]{a_n} \\le r\\]\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774\uac83\uc740 \uc55e\uc5d0\uc11c \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc744 \uc99d\uba85\ud560 \ub54c\uc640 \uac19\uc740 \uc0c1\ud669\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uacfc \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

        \\(\\rho > 1\\)\uc778 \uacbd\uc6b0, \\(\\rho\\)\uc640 \\(1\\) \uc0ac\uc774\uc5d0\uc11c \uc2e4\uc218 \\(r\\)\uc744 \ud0dd\ud560 \uc218 \uc788\ub2e4. [\\(\\rho = \\infty\\)\uc77c \uc218 \uc788\uae30 \ub54c\ubb38\uc5d0 \uc774\ub7ec\ud55c \uc791\uc5c5\uc774 \ud544\uc694\ud558\ub2e4.] \uadf8\ub7ec\uba74 \\(\\left\\{ \\sqrt[n]{a_n}\\right\\}\\)\uc758 \ubd80\ubd84\uc218\uc5f4 \uc911\uc5d0\uc11c \\(r\\)\ubcf4\ub2e4 \ud070 \uac12\uc5d0 \uc218\ub834\ud558\ub294 \uac83\uc774 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7ec\ud55c \ubd80\ubd84\uc218\uc5f4\uc740 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c \\(\\left\\{ a_n \\right\\}\\) \ub610\ud55c \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n

        \uc774\ub85c\uc368 \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

        \n

        \uc815\ub9ac 2.2.7. (\uc81c\uacf1\uadfc \uc0c1\uadf9\ud55c \ud310\uc815\ubc95, Revised Root Test)<\/span><\/p>\n

        \uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0
        \n\\[\\varlimsup_{n\\rightarrow\\infty} \\sqrt[n]{a_n} = \\rho\\]
        \n\uc774\uba70 \\(\\rho\\)\uac00 \uc0c1\uc218\ub77c\uace0 \ud558\uc790.<\/p>\n

          \n
        1. \ub9cc\uc57d \\(\\rho < 1\\)\uc774\uba74 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/li>\n
        2. \ub9cc\uc57d \\(\\rho > 1\\)\uc774\uba74 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n
        3. \ub9cc\uc57d \\(\\rho = 1\\)\uc774\uba74 \uc774 \ud310\uc815\ubc95\uc73c\ub85c \\(\\sum_{n=1}^{\\infty} a_n\\)\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \ud310\uc815\ud560 \uc218 \uc5c6\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

          \uc704 \uc815\ub9ac\uc758 \uc774\ub984\uc744 \u2018\uc81c\uacf1\uadfc \uc0c1\uadf9\ud55c \ud310\uc815\ubc95\u2019\uc774\ub77c\uace0 \ubd99\uc600\uc9c0\ub9cc, \uc0ac\uc2e4 \uadf8\ub0e5 \u2018\uc81c\uacf1\uadfc \ud310\uc815\ubc95\u2019\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

          \n

          \ubcf4\uae30 2.2.10.<\/span>
          \n\uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc774 \ub2e4\uc74c\uacfc \uac19\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.
          \n\\[a_n : \\quad \\frac{1}{2} ,\\,\\, \\frac{1}{1^1} ,\\,\\, \\frac{1}{2^2} ,\\,\\, \\frac{1}{2^2} ,\\,\\, \\frac{1}{2^3} ,\\,\\, \\frac{1}{3^3} ,\\,\\, \\frac{1}{2^4} ,\\,\\, \\frac{1}{4^4} ,\\,\\, \\cdots\\]
          \n\uc989
          \n\\[a_{2n} = \\frac{1}{n^n} ,\\quad a_{2n-1} = \\frac{1}{2^n}\\]
          \n\uc774\ub2e4. \uadf8\ub7ec\uba74
          \n\\[\\lim_{n\\rightarrow\\infty} \\sqrt[2n]{a_{2n}} = \\lim_{n\\rightarrow\\infty} \\frac{1}{\\sqrt{n}} = 0\\]
          \n\uadf8\ub9ac\uace0
          \n\\[\\lim_{n\\rightarrow\\infty} \\sqrt[2n-1]{a_{2n-1}} = \\lim_{n\\rightarrow\\infty} \\frac{1}{2^{\\frac{n}{2n-1}}} = \\frac{1}{\\sqrt{2}}\\]
          \n\uc774\ub2e4. \\(\\left\\{ a_n \\right\\}\\)\uc758 \uac01 \ud56d\uc740 \\(\\left\\{ a_{2n} \\right\\}\\) \ub610\ub294 \\(\\left\\{ a_{2n-1}\\right\\}\\)\uc5d0 \uc18d\ud55c\ub2e4. \ub530\ub77c\uc11c \\(\\left\\{ \\sqrt[n]{a_n}\\right\\}\\)\uc758 \uc0c1\uadf9\ud55c\uc740 \\(\\frac{1}{\\sqrt{2}}\\)\uc744 \ub118\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n

          \uadf8\ub7ec\ubbc0\ub85c \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(\\sum_{n=1}^{\\infty} a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4.\n<\/p>\n<\/div>\n

          <\/p>\n

          <\/p>\n

          \n