{"id":9491,"date":"2025-10-20T18:58:30","date_gmt":"2025-10-20T09:58:30","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9491"},"modified":"2026-09-27T14:57:44","modified_gmt":"2026-09-27T05:57:44","slug":"ch07-infinite-series","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-analysis\/ch07-infinite-series\/","title":{"rendered":"\ubb34\ud55c\uae09\uc218"},"content":{"rendered":"
<\/p>\n

<\/p>\n

\uc55e \uc7a5\ub4e4\uc5d0\uc11c\ub294 \uc218\uc5f4\uc758 \uc218\ub834\uacfc \ud568\uc218\uc758 \uadf9\ud55c, \ubbf8\ubd84, \uc801\ubd84\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc218\uc5f4\uc758 \ubd80\ubd84\ud569\uc744 \ub2e4\uc2dc \ud558\ub098\uc758 \uc218\uc5f4\ub85c \ubcf4\uc544 \ubb34\ud55c\uae09\uc218\ub97c \uc815\uc758\ud558\uace0, \uc218\uc5f4\uc758 \uc218\ub834 \uc774\ub860\uacfc \uc5ec\ub7ec \uc218\ub834 \ud310\uc815\ubc95\uc744 \uc804\uac1c\ud55c\ub2e4. \ud2b9\ubcc4\ud55c \uc5b8\uae09\uc774 \uc5c6\uc73c\uba74 \uc774 \uc7a5\uc758 \uc218\uc5f4\uc740 \uc2e4\uc218\uc5f4\ub85c \uac00\uc815\ud55c\ub2e4.<\/p>\n

\ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834\uacfc \ubc1c\uc0b0<\/h3>\n

\uc218\uc5f4 \\(\\{a_n\\}\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[
\nS_N=\\sum_{n=1}^N a_n
\n\\]
\n\uc744 \\(N\\)\ubc88\uc9f8 \ubd80\ubd84\ud569<\/span>(partial sum)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uae30\ud638
\n\\[
\n\\sum_{n=1}^{\\infty}a_n
\n\\]
\n\uc744 \\(\\{a_n\\}\\)\uc758 \ubb34\ud55c\uae09\uc218<\/span>(infinite series)\ub77c\uace0 \ubd80\ub978\ub2e4. \ubd80\ubd84\ud569 \uc218\uc5f4 \\(\\{S_N\\}\\)\uc774 \uc218\ub834\ud558\uba74 \uc774 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud55c\ub2e4(converge)\uace0 \ub9d0\ud558\uace0, \uadf8 \uadf9\ud55c\uc744 \uae09\uc218\uc758 \ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \\(S_N\\to S\\)\uc774\uba74
\n\\[
\n\\sum_{n=1}^{\\infty}a_n=S
\n\\]
\n\uc73c\ub85c \uc4f4\ub2e4. \ubd80\ubd84\ud569 \uc218\uc5f4\uc774 \uc218\ub834\ud558\uc9c0 \uc54a\uc73c\uba74 \uae09\uc218\uac00 \ubc1c\uc0b0\ud55c\ub2e4\uace0 \ub9d0\ud55c\ub2e4. \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty}a_n\\)\uc744 \uac04\ub2e8\ud788 \\(\\sum a_n\\)\uc73c\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n

\ud56d\ubc88\ud638\uac00 \ubc18\ub4dc\uc2dc \\(1\\)\ubd80\ud130 \uc2dc\uc791\ud560 \ud544\uc694\ub294 \uc5c6\ub2e4. \uc608\ub97c \ub4e4\uc5b4
\n\\[
\n\\sum_{n=2}^{\\infty}a_n,\\quad
\n\\sum_{k=0}^{\\infty}b_k,\\quad
\n\\sum_{j=4}^{\\infty}c_j
\n\\]
\n\ub3c4 \ubaa8\ub450 \uac19\uc740 \ubc29\uc2dd\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \uc720\ud55c\uac1c\uc758 \ud56d\uc744 \ub354\ud558\uac70\ub098 \ube7c\ub294 \uac83\uc740 \uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\uc5d0 \uc601\ud5a5\uc744 \uc8fc\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n

\uac00\uc7a5 \uae30\ubcf8\uc801\uc778 \uc608\ub294 \uae30\ud558\uae09\uc218\uc774\ub2e4.<\/p>\n

\n

\uc815\ub9ac 7.1. (\uae30\ud558\uae09\uc218)<\/span><\/p>\n

\uc2e4\uc218 \\(r\\)\uc5d0 \ub300\ud558\uc5ec \uae30\ud558\uae09\uc218 \\(\\sum_{n=0}^{\\infty}r^n\\)\uc740 \\(|r|<1\\)\uc77c \ub54c \uadf8\ub9ac\uace0 \uadf8\ub54c\uc5d0\ub9cc \uc218\ub834\ud55c\ub2e4. \uc774 \uacbd\uc6b0
\n\\[
\n\\sum_{n=0}^{\\infty}r^n=\\frac1{1-r}.
\n\\]<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/span>
\n\\(r\\ne1\\)\uc77c \ub54c \\(N\\)\ubc88\uc9f8 \ubd80\ubd84\ud569\uc740
\n\\[
\n1+r+\\cdots+r^N=\\frac{1-r^{N+1}}{1-r}
\n\\]
\n\uc774\ub2e4. \\(|r|<1\\)\uc774\uba74 \\(r^{N+1}\\to0\\)\uc774\ubbc0\ub85c \ubd80\ubd84\ud569\uc740 \\(1\/(1-r)\\)\ub85c \uc218\ub834\ud55c\ub2e4. \ubc18\ub300\ub85c \\(|r|\\ge1\\)\uc774\uba74 \uc77c\ubc18\ud56d \\(r^n\\)\uc774 \\(0\\)\uc73c\ub85c \uc218\ub834\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c \uae09\uc218\ub294 \uc218\ub834\ud560 \uc218 \uc5c6\ub2e4.<\/span><\/p>\n<\/div>\n

\uae09\uc218\uac00 \uc218\ub834\ud558\uba74 \uc77c\ubc18\ud56d\uc740 \ubc18\ub4dc\uc2dc \\(0\\)\uc73c\ub85c \uc218\ub834\ud55c\ub2e4. \uc2e4\uc81c\ub85c \\(S_n=\\sum_{k=1}^n a_k\\)\ub77c\uace0 \ud558\uba74
\n\\[
\na_n=S_n-S_{n-1}\\longrightarrow S-S=0.
\n\\]
\n\uadf8\ub7ec\ub098 \uc5ed\uc740 \uc77c\ubc18\uc801\uc73c\ub85c \ucc38\uc774 \uc544\ub2c8\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \\(1\/n\\to0\\)\uc774\uc9c0\ub9cc \uc870\ud654\uae09\uc218 \\(\\sum_{n=1}^{\\infty}1\/n\\)\uc740 \ub4a4\uc758 \\(p\\)-\uae09\uc218 \ud310\uc815\uc5d0\uc11c \ubc1c\uc0b0\ud568\uc744 \ud655\uc778\ud55c\ub2e4.<\/p>\n

\uc2e4\uc218\uc5f4\uc758 \ucf54\uc2dc \ud310\uc815\ubc95\uc744 \ubd80\ubd84\ud569 \uc218\uc5f4\uc5d0 \uc801\uc6a9\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

\n

\uc815\ub9ac 7.2. (\ubb34\ud55c\uae09\uc218\uc758 \ucf54\uc2dc \ud310\uc815\ubc95)<\/span><\/p>\n

\ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty}a_n\\)\uc774 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \uc5b4\ub5a4 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(m>n\\ge N\\)\uc774\uba74
\n\\[
\n\\left|\\sum_{k=n+1}^m a_k\\right|<\\varepsilon
\n\\]
\n\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/span>
\n\ubd80\ubd84\ud569\uc744 \\(S_n=\\sum_{k=1}^n a_k\\)\ub77c\uace0 \ud558\uba74
\n\\[
\nS_m-S_n=\\sum_{k=n+1}^m a_k.
\n\\]
\n\ub530\ub77c\uc11c \uc704 \uc870\uac74\uc740 \ubd80\ubd84\ud569 \uc218\uc5f4 \\(\\{S_n\\}\\)\uc774 \ucf54\uc2dc \uc218\uc5f4\uc774\ub77c\ub294 \uc870\uac74\uacfc \uc815\ud655\ud788 \uac19\ub2e4. \uc2e4\uc218\uc5f4\uc774 \ucf54\uc2dc \uc218\uc5f4\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uadf8 \uc218\uc5f4\uc774 \uc218\ub834\ud558\ub294 \uac83\uc774\ubbc0\ub85c, \ubc14\ub77c\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

\ubb34\ud55c\uae09\uc218 \\(\\sum|a_n|\\)\uc774 \uc218\ub834\ud560 \ub54c \u201c\\(\\sum a_n\\)\uc774 \uc808\ub300\uc218\ub834<\/span>\ud55c\ub2e4(converges absolutely)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ub610\ud55c \\(\\sum a_n\\)\uc774 \uc218\ub834\ud558\uc9c0\ub9cc \uc808\ub300\uc218\ub834\ud558\uc9c0 \uc54a\uc744 \ub54c \u201c\\(\\sum a_n\\)\uc774 \uc870\uac74\uc218\ub834<\/span>\ud55c\ub2e4(converges conditionally)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n

\n

\uc815\ub9ac 7.3. (\uc808\ub300\uc218\ub834\uacfc \uc218\ub834)<\/span><\/p>\n

\uc808\ub300\uc218\ub834\ud558\ub294 \ubb34\ud55c\uae09\uc218\ub294 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/span>
\n\\(\\sum|a_n|\\)\uc774 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \ucf54\uc2dc \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud574 \ucda9\ubd84\ud788 \ud070 \\(m>n\\)\uc774\uba74
\n\\[
\n\\sum_{k=n+1}^m|a_k|<\\varepsilon.
\n\\]
\n\ub530\ub77c\uc11c
\n\\[
\n\\left|\\sum_{k=n+1}^m a_k\\right|
\n\\le\\sum_{k=n+1}^m|a_k|
\n<\\varepsilon.
\n\\]
\n\uc815\ub9ac 7.2\uc5d0 \uc758\ud574 \\(\\sum a_n\\)\uc740 \uc218\ub834\ud55c\ub2e4.<\/span><\/p>\n<\/div>\n

\uc5ed\uc740 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \uad50\ub300\uc870\ud654\uae09\uc218 \\(\\sum_{n=1}^{\\infty}(-1)^{n+1}\/n\\)\uc740 \ub4a4\uc758 \uad50\ub300\uae09\uc218 \ud310\uc815\ubc95\uc73c\ub85c \uc218\ub834\ud558\uc9c0\ub9cc \uc808\ub300\uc218\ub834\ud558\uc9c0 \uc54a\uc74c\uc744 \ud655\uc778\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\n

\ubb38\uc81c 7.1.<\/span>
\n\ubb34\ud55c\uae09\uc218 \\(\\sum a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud558\uace0 \uadf8 \ud569\uc774 \\(S\\)\ub77c\uace0 \ud558\uc790. \\(R_N=S-\\sum_{n=1}^N a_n\\)\uc774\ub77c\uace0 \ud560 \ub54c
\n\\[
\n|R_N|\\le\\sum_{n=N+1}^{\\infty}|a_n|
\n\\]
\n\uc784\uc744 \ubcf4\uc774\uace0, \uc774\ub97c \uc774\uc6a9\ud558\uc5ec \\(R_N\\to0\\)\uc784\uc744 \ub2e4\uc2dc \ud655\uc778\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

\ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834 \ud310\uc815\ubc95<\/h3>\n

\ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud558\ub294\uc9c0 \uc5ec\ubd80\ub97c \ud310\ubcc4\ud558\ub294 \uacf5\uc2dd\uc744 \ud310\uc815\ubc95<\/span>(test)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\ube44\uad50 \ud310\uc815\ubc95<\/h4>\n

\ubb34\ud55c\uae09\uc218 \\(\\sum a_n\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc77c \ub54c, \uc989 \\(a_n\\ge0\\)\uc77c \ub54c \\(\\sum a_n\\)\uc744 \uc591\ud56d\uae09\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \uc591\ud56d\uae09\uc218\uc758 \ubd80\ubd84\ud569 \uc218\uc5f4\uc740 \ub2e8\uc870\uc99d\uac00\ud55c\ub2e4. \ub530\ub77c\uc11c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud574 \uc591\ud56d\uae09\uc218\uac00 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ubd80\ubd84\ud569 \uc218\uc5f4\uc774 \uc704\ub85c \uc720\uacc4\uc778 \uac83\uc774\ub2e4.<\/p>\n

\n

\uc815\ub9ac 7.4. (\uc591\ud56d\uae09\uc218\uc758 \ube44\uad50 \ud310\uc815\ubc95)<\/span><\/p>\n

\ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(0\\le a_n\\le b_n\\)\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\sum b_n\\)\uc774 \uc218\ub834\ud558\uba74 \\(\\sum a_n\\)\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/span>
\n\uc720\ud55c\uac1c\uc758 \ucc98\uc74c \ud56d\uc740 \uc218\ub834 \uc5ec\ubd80\uc5d0 \uc601\ud5a5\uc744 \uc8fc\uc9c0 \uc54a\uc73c\ubbc0\ub85c \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(0\\le a_n\\le b_n\\)\uc774\ub77c\uace0 \uac00\uc815\ud574\ub3c4 \ub41c\ub2e4. \ubd80\ubd84\ud569\uc744 \uac01\uac01 \\(A_n\\), \\(B_n\\)\uc774\ub77c\uace0 \ud558\uba74 \\(0\\le A_n\\le B_n\\)\uc774\ub2e4. \\(\\{B_n\\}\\)\uc774 \uc218\ub834\ud558\ubbc0\ub85c \uc704\ub85c \uc720\uacc4\uc774\uace0, \ub530\ub77c\uc11c \ub2e8\uc870\uc99d\uac00\uc218\uc5f4 \\(\\{A_n\\}\\)\ub3c4 \uc704\ub85c \uc720\uacc4\uc774\ub2e4. \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud574 \\(\\{A_n\\}\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/span><\/p>\n<\/div>\n

\uc774 \uc815\ub9ac\uc758 \ub300\uc6b0\ub97c \uc0ac\uc6a9\ud558\uba74, \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0\uc11c \\(0\\le b_n\\le a_n\\)\uc774\uace0 \\(\\sum b_n\\)\uc774 \ubc1c\uc0b0\ud560 \ub54c \\(\\sum a_n\\)\ub3c4 \ubc1c\uc0b0\ud568\uc744 \uc54c \uc218 \uc788\ub2e4.<\/p>\n

\n

\uc815\ub9ac 7.5. (\uc591\ud56d\uae09\uc218\uc758 \uadf9\ud55c\ube44\uad50 \ud310\uc815\ubc95)<\/span><\/p>\n

\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n\\ge0\\), \\(b_n>0\\)\uc774\uace0
\n\\[
\n\\lim_{n\\to\\infty}\\frac{a_n}{b_n}=L
\n\\]
\n\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

    \n
  1. \\(0<L<\\infty\\)\uc774\uba74 \ub450 \ubb34\ud55c\uae09\uc218\ub294 \ud568\uaed8 \uc218\ub834\ud558\uac70\ub098 \ud568\uaed8 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n
  2. \\(L=0\\)\uc774\uace0 \\(\\sum b_n\\)\uc774 \uc218\ub834\ud558\uba74 \\(\\sum a_n\\)\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/li>\n
  3. \\(L=\\infty\\)\uc774\uace0 \\(\\sum b_n\\)\uc774 \ubc1c\uc0b0\ud558\uba74 \\(\\sum a_n\\)\ub3c4 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
    \n

    \uc99d\uba85<\/span>
    \n\uba3c\uc800 \\(0<L<\\infty\\)\ub77c\uace0 \ud558\uc790. \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n\\frac L2<\\frac{a_n}{b_n}<\\frac{3L}{2}
    \n\\]
    \n\uc774\ubbc0\ub85c
    \n\\[
    \n\\frac L2 b_n<a_n<\\frac{3L}{2}b_n.
    \n\\]
    \n\ube44\uad50 \ud310\uc815\ubc95\uc744 \uc591\ubc29\ud5a5\uc73c\ub85c \uc801\uc6a9\ud558\uba74 \ub450 \uae09\uc218\uc758 \uc218\ub834\uc131\uc774 \uac19\ub2e4.<\/p>\n

    \\(L=0\\)\uc774\uba74 \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0\uc11c \\(a_n\/b_n<1\\), \uc989 \\(a_n<b_n\\)\uc774\ubbc0\ub85c \\(\\sum b_n\\)\uc758 \uc218\ub834\uc73c\ub85c\ubd80\ud130 \\(\\sum a_n\\)\uc758 \uc218\ub834\uc744 \uc5bb\ub294\ub2e4. \ub9c8\uc9c0\ub9c9\uc73c\ub85c \\(L=\\infty\\)\uc774\uba74 \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0\uc11c \\(a_n>b_n\\)\uc774\ub2e4. \uc774\ub54c \\(\\sum a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uba74 \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud574 \\(\\sum b_n\\)\ub3c4 \uc218\ub834\ud558\uc5ec \uac00\uc815\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

    \ub450 \uc218\uc5f4\uc744 \uc9c1\uc811 \ube44\uad50\ud558\ub294 \ub300\uc2e0 \ud568\uc218\uc758 \uc801\ubd84\uacfc \ube44\uad50\ud560 \uc218\ub3c4 \uc788\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 7.6. (\uc801\ubd84 \ud310\uc815\ubc95)<\/span><\/p>\n

    \ud568\uc218 \\(f\\)\uac00 \\([1,\\infty)\\)\uc5d0\uc11c \uac10\uc18c\ud558\uace0 \\(f(x)\\ge0\\)\uc774\uba70 \\(a_n=f(n)\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^{\\infty}a_n\\)\uacfc \uc774\uc0c1\uc801\ubd84 \\(\\int_1^\\infty f(x)\\,dx\\)\ub294 \ud568\uaed8 \uc218\ub834\ud558\uac70\ub098 \ud568\uaed8 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/span>
    \n\\(f\\)\uac00 \uac10\uc18c\ud558\ubbc0\ub85c \uc790\uc5f0\uc218 \\(n\\ge1\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \nf(n+1)\\le\\int_n^{n+1}f(x)\\,dx\\le f(n)
    \n\\]
    \n\uc774\ub2e4. \ub530\ub77c\uc11c \ubaa8\ub4e0 \uc790\uc5f0\uc218 \\(N\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n\\int_1^{N+1}f(x)\\,dx
    \n\\le\\sum_{n=1}^N f(n)
    \n\\le f(1)+\\int_1^N f(x)\\,dx.
    \n\\]
    \n\uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\uba74 \uc624\ub978\ucabd \ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud574 \uc591\ud56d\uae09\uc218\uc758 \ubd80\ubd84\ud569\uc774 \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c \uae09\uc218\uac00 \uc218\ub834\ud55c\ub2e4. \ubc18\ub300\ub85c \uae09\uc218\uac00 \uc218\ub834\ud558\uba74 \uc67c\ucabd \ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud574 \\(\\int_1^{N+1}f\\)\uac00 \uc704\ub85c \uc720\uacc4\uc774\ub2e4. \uc774 \uc801\ubd84\uac12\ub4e4\uc740 \\(N\\)\uc5d0 \ub300\ud574 \ub2e8\uc870\uc99d\uac00\ud558\ubbc0\ub85c \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\uace0, \ub530\ub77c\uc11c \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud55c\ub2e4.<\/span><\/p>\n<\/div>\n

    \n

    \uc815\ub9ac 7.7. (\\(p\\)-\uae09\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n

    \uc2e4\uc218 \\(p\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n\\sum_{n=1}^{\\infty}\\frac1{n^p}
    \n\\]
    \n\ub294 \\(p>1\\)\uc77c \ub54c \uadf8\ub9ac\uace0 \uadf8\ub54c\uc5d0\ub9cc \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/span>
    \n\\(p>0\\)\uc77c \ub54c \\(f(x)=x^{-p}\\)\ub294 \\([1,\\infty)\\)\uc5d0\uc11c \uac10\uc18c\ud558\uace0 \uc74c\uc774 \uc544\ub2c8\ub2e4. \uc801\ubd84 \ud310\uc815\ubc95\uacfc
    \n\\[
    \n\\int_1^\\infty x^{-p}\\,dx
    \n\\]
    \n\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \ube44\uad50\ud558\uba74 \\(p>1\\)\uc77c \ub54c \uc815\ud655\ud788 \uc218\ub834\ud55c\ub2e4. \\(p\\le0\\)\uc774\uba74 \uc77c\ubc18\ud56d \\(1\/n^p\\)\uac00 \\(0\\)\uc73c\ub85c \uc218\ub834\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c \uae09\uc218\ub294 \ubc1c\uc0b0\ud55c\ub2e4.<\/span><\/p>\n<\/div>\n

    \n

    \ubb38\uc81c 7.2.<\/span>
    \n\\(\\{a_n\\}\\)\uc774 \uac10\uc18c\ud558\ub294 \uc218\uc5f4\uc774\uace0 \\(a_n\\ge0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub450 \ubb34\ud55c\uae09\uc218
    \n\\[
    \n\\sum_{n=1}^{\\infty}a_n,\\quad
    \n\\sum_{k=0}^{\\infty}2^k a_{2^k}
    \n\\]
    \n\ub294 \ud568\uaed8 \uc218\ub834\ud558\uac70\ub098 \ud568\uaed8 \ubc1c\uc0b0\ud568\uc744 \ubcf4\uc774\uc2dc\uc624. \uc774 \ud310\uc815\ubc95\uc744 \ucf54\uc2dc \uc751\uc9d1 \ud310\uc815\ubc95<\/span>(Cauchy condensation test)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n

    \n

    \ubb38\uc81c 7.3.<\/span>
    \n\ub2e4\uc74c \ubb34\ud55c\uae09\uc218\ub97c \ud310\uc815\ud558\uc2dc\uc624.<\/p>\n

      \n
    1. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac1n\\)<\/li>\n
    2. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac1{\\sqrt n}\\)<\/li>\n
    3. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac1{n^2}\\)<\/li>\n
    4. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac n{3^n}\\)<\/li>\n
    5. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac{n+1}{n^3-n+1}\\)<\/li>\n
    6. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac{n^2+n}{n^3-2n+2}\\)<\/li>\n
    7. \\(\\displaystyle\\sum_{n=2}^{\\infty}\\frac1{n\\ln n}\\)<\/li>\n
    8. \\(\\displaystyle\\sum_{n=2}^{\\infty}\\frac1{n(\\ln n)^2}\\)<\/li>\n
    9. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac{n!}{n^n}\\)<\/li>\n<\/ol>\n<\/div>\n

      \ube44 \ud310\uc815\ubc95\uacfc \uc81c\uacf1\uadfc \ud310\uc815\ubc95<\/h4>\n

      \ubb34\ud55c\uae09\uc218\ub97c \ud310\uc815\ud560 \ub54c \ub2e4\ub978 \ubb34\ud55c\uae09\uc218\uc640 \ube44\uad50\ud558\ub294 \uac83\uc774 \uc544\ub2c8\ub77c \uc790\uc2e0\uc758 \uc778\uc811\ud55c \ud56d\uacfc \ube44\uad50\ud558\ub294 \ubc29\ubc95\uc774 \uc788\ub2e4.<\/p>\n

      \n

      \uc815\ub9ac 7.8. (\ube44 \ud310\uc815\ubc95)<\/span><\/p>\n

      \uc218\uc5f4 \\(\\{a_n\\}\\)\uc774 \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0\uc11c \\(a_n\\ne0\\)\uc774\uace0
      \n\\[
      \n\\lim_{n\\to\\infty}\\left|\\frac{a_{n+1}}{a_n}\\right|=L\\in[0,\\infty]
      \n\\]
      \n\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

        \n
      1. \\(L<1\\)\uc774\uba74 \\(\\sum a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud55c\ub2e4.<\/li>\n
      2. \\(L>1\\) \ub610\ub294 \\(L=\\infty\\)\uc774\uba74 \\(\\sum a_n\\)\uc774 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n
      3. \\(L=1\\)\uc774\uba74 \uc774 \ud310\uc815\ubc95\ub9cc\uc73c\ub85c\ub294 \\(\\sum a_n\\)\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \uacb0\uc815\ud560 \uc218 \uc5c6\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
        \n

        \uc99d\uba85<\/span>
        \n\\(L<1\\)\uc774\ub77c\uace0 \ud558\uc790. \\(L<r<1\\)\uc778 \\(r\\)\uc744 \ud0dd\ud558\uba74 \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0\uc11c
        \n\\[
        \n\\left|\\frac{a_{n+1}}{a_n}\\right|<r.
        \n\\]
        \n\ub530\ub77c\uc11c \uc5b4\ub5a4 \\(N\\)\uacfc \uc0c1\uc218 \\(C>0\\)\uc5d0 \ub300\ud558\uc5ec \\(|a_{N+k}|\\le Cr^k\\)\uc774\ub2e4. \uc815\ub9ac 7.1\uacfc \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud574 \\(\\sum|a_n|\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n

        \\(L>1\\)\uc774\uba74 \\(1<r<L\\)\uc778 \\(r\\)\uc744 \ud0dd\ud560 \uc218 \uc788\uace0, \\(L=\\infty\\)\uc778 \uacbd\uc6b0\uc5d0\ub3c4 \uc784\uc758\uc758 \\(r>1\\)\uc744 \ud0dd\ud560 \uc218 \uc788\ub2e4. \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0\uc11c
        \n\\[
        \n|a_{n+1}|>r|a_n|
        \n\\]
        \n\uc774\ubbc0\ub85c \\(a_n\\)\uc740 \\(0\\)\uc73c\ub85c \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \ub530\ub77c\uc11c \uae09\uc218\ub294 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

        \ub9c8\uc9c0\ub9c9\uc73c\ub85c \\(\\sum1\/n\\)\uacfc \\(\\sum1\/n^2\\)\uc5d0\uc11c\ub294 \ubaa8\ub450 \uc704 \ube44\uc758 \uadf9\ud55c\uc774 \\(1\\)\uc774\uc9c0\ub9cc, \uccab \uae09\uc218\ub294 \ubc1c\uc0b0\ud558\uace0 \ub458\uc9f8 \uae09\uc218\ub294 \uc218\ub834\ud55c\ub2e4. \ub530\ub77c\uc11c \\(L=1\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \uacb0\ub860\uc744 \ub0b4\ub9b4 \uc218 \uc5c6\ub2e4.<\/span><\/p>\n<\/div>\n

        3\uc7a5<\/a>\uc5d0\uc11c\ub294 \uc720\uacc4\uc218\uc5f4\uc5d0 \ub300\ud574 \uc0c1\uadf9\ud55c\uc744 \uc815\uc758\ud588\ub2e4. \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc5d0\uc11c\ub294 \ube44\uc74c\uc218 \uc218\uc5f4 \\(c_n\\)\uc5d0 \ub300\ud558\uc5ec \uac19\uc740 \uaf2c\ub9ac \uc0c1\ud55c\uc758 \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\ub418, \uac12 \\(+\\infty\\)\ub3c4 \ud5c8\uc6a9\ud55c\ub2e4. \uc989
        \n\\[
        \n\\varlimsup_{n\\to\\infty}c_n
        \n=\\inf_{N\\ge1}\\sup_{n\\ge N}c_n\\in[0,\\infty].
        \n\\]<\/p>\n

        \n

        \uc815\ub9ac 7.9. (\uc81c\uacf1\uadfc \ud310\uc815\ubc95)<\/span><\/p>\n

        \uc218\uc5f4 \\(\\{a_n\\}\\)\uc5d0 \ub300\ud558\uc5ec
        \n\\[
        \nL=\\varlimsup_{n\\to\\infty}\\sqrt[n]{|a_n|}\\in[0,\\infty]
        \n\\]
        \n\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

          \n
        1. \\(L<1\\)\uc774\uba74 \\(\\sum a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud55c\ub2e4.<\/li>\n
        2. \\(L>1\\) \ub610\ub294 \\(L=\\infty\\)\uc774\uba74 \\(\\sum a_n\\)\uc774 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n
        3. \\(L=1\\)\uc774\uba74 \uc774 \ud310\uc815\ubc95\ub9cc\uc73c\ub85c\ub294 \\(\\sum a_n\\)\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \uacb0\uc815\ud560 \uc218 \uc5c6\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
          \n

          \uc99d\uba85<\/span>
          \n\\(L<1\\)\uc774\ub77c\uace0 \ud558\uc790. \\(L<r<1\\)\uc778 \\(r\\)\uc744 \ud0dd\ud558\uba74 \uc0c1\uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud574 \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0\uc11c
          \n\\[
          \n\\sqrt[n]{|a_n|}<r,
          \n\\]
          \n\uc989 \\(|a_n|<r^n\\)\uc774\ub2e4. \ub530\ub77c\uc11c \uae30\ud558\uae09\uc218\uc640 \ube44\uad50\ud558\uba74 \\(\\sum|a_n|\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n

          \\(L>1\\)\uc774\uba74 \\(1<r<L\\)\uc778 \\(r\\)\uc744 \ud0dd\ud560 \uc218 \uc788\uace0, \\(L=\\infty\\)\uc774\uba74 \uc784\uc758\uc758 \\(r>1\\)\uc744 \ud0dd\ud560 \uc218 \uc788\ub2e4. \uc0c1\uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud574
          \n\\[
          \n\\sqrt[n]{|a_n|}>r
          \n\\]
          \n\uc778 \\(n\\)\uc774 \ubb34\ud55c\ud788 \ub9ce\ub2e4. \ub530\ub77c\uc11c \uadf8\ub7f0 \\(n\\)\uc5d0 \ub300\ud574 \\(|a_n|>r^n>1\\)\uc774\ubbc0\ub85c \\(a_n\\not\\to0\\)\uc774\uace0 \uae09\uc218\ub294 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

          \\(\\sum1\/n\\)\uacfc \\(\\sum1\/n^2\\)\uc5d0\uc11c\ub294 \ubaa8\ub450 \\(n\\)\uc81c\uacf1\uadfc\uc758 \uadf9\ud55c\uc774 \\(1\\)\uc774\ubbc0\ub85c \\(L=1\\)\uc774\uc9c0\ub9cc \uc218\ub834\uc131\uc740 \uc11c\ub85c \ub2e4\ub974\ub2e4.<\/span><\/p>\n<\/div>\n

          \uad50\ub300\uae09\uc218 \ud310\uc815\ubc95<\/h4>\n

          \uc218\uc5f4 \\(\\{a_n\\}\\)\uc5d0 \ub300\ud558\uc5ec \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc778 \uc218\uc5f4 \\(\\{u_n\\}\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(a_n=(-1)^n u_n\\) \ub610\ub294 \\(a_n=(-1)^{n+1}u_n\\)\uaf34\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\uc744 \ub54c, \\(\\{a_n\\}\\)\uc744 \uad50\ub300\uc218\uc5f4<\/span>(alternating sequence)\uc774\ub77c\uace0 \ubd80\ub974\uace0 \ubb34\ud55c\uae09\uc218 \\(\\sum a_n\\)\uc744 \uad50\ub300\uae09\uc218<\/span>(alternating series)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

          \n

          \uc815\ub9ac 7.10. (\uad50\ub300\uae09\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n

          \uc218\uc5f4 \\(\\{u_n\\}\\)\uc774 \\(u_n\\ge u_{n+1}\\ge0\\)\uc744 \ub9cc\uc871\ud55c\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \ubb34\ud55c\uae09\uc218
          \n\\[
          \n\\sum_{n=1}^{\\infty}(-1)^n u_n
          \n\\]
          \n\uc774 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(u_n\\to0\\)\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

          \n

          \uc99d\uba85<\/span>
          \n\uae09\uc218\uac00 \uc218\ub834\ud558\uba74 \uc77c\ubc18\ud56d \\((-1)^n u_n\\)\uc774 \\(0\\)\uc73c\ub85c \uc218\ub834\ud558\ubbc0\ub85c \\(u_n\\to0\\)\uc774\ub2e4.<\/p>\n

          \uc5ed\uc73c\ub85c \\(u_n\\to0\\)\uc774\ub77c\uace0 \ud558\uc790. \ubd80\ubd84\ud569\uc744
          \n\\[
          \ns_n=\\sum_{k=1}^n(-1)^k u_k
          \n\\]
          \n\ub77c\uace0 \ud558\uc790. \\(u_n\\)\uc758 \ub2e8\uc870\uc131\uc5d0 \uc758\ud574 \\(\\{s_{2n-1}\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud558\uace0 \\(\\{s_{2n}\\}\\)\uc740 \ub2e8\uc870\uac10\uc18c\ud558\uba70
          \n\\[
          \ns_{2n-1}\\le s_{2n}
          \n\\]
          \n\uc774\ub2e4. \ub530\ub77c\uc11c \ud640\uc218 \ubc88\uc9f8 \ubd80\ubd84\ud569 \uc218\uc5f4\uc740 \uc704\ub85c \uc720\uacc4\uc774\uace0 \uc9dd\uc218 \ubc88\uc9f8 \ubd80\ubd84\ud569 \uc218\uc5f4\uc740 \uc544\ub798\ub85c \uc720\uacc4\uc774\ubbc0\ub85c \ub458 \ub2e4 \uc218\ub834\ud55c\ub2e4. \ub610\ud55c
          \n\\[
          \ns_{2n}-s_{2n-1}=u_{2n}\\longrightarrow0
          \n\\]
          \n\uc774\ubbc0\ub85c \ub450 \ubd80\ubd84\uc218\uc5f4\uc758 \uadf9\ud55c\uc740 \uac19\ub2e4. \ub530\ub77c\uc11c \uc804\uccb4 \ubd80\ubd84\ud569 \uc218\uc5f4 \\(\\{s_n\\}\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/span><\/p>\n<\/div>\n

          \n

          \ubb38\uc81c 7.4.<\/span>
          \n\ub2e4\uc74c \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud558\ub3c4\ub85d \ud558\ub294 \\(x\\)\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n

            \n
          1. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac1n x^n\\)<\/li>\n
          2. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac n{n+1}x^n\\)<\/li>\n
          3. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac1{n(n+1)}x^n\\)<\/li>\n
          4. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac1{n5^n}x^n\\)<\/li>\n
          5. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac1{n(n+1)(n+2)}x^{2n}\\)<\/li>\n
          6. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac1{(\\ln(n+1))^2}x^{n+1}\\)<\/li>\n
          7. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac1{1+n^3}x^n\\)<\/li>\n
          8. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac1{n^2}(x-4)^n\\)<\/li>\n
          9. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac1{5^n}(3x-2)^n\\)<\/li>\n<\/ol>\n<\/div>\n
            \n

            \ubb38\uc81c 7.5.<\/span>
            \n\uc218\uc5f4 \\(\\{u_n\\}\\)\uc774 \\(u_n\\ge u_{n+1}\\ge0\\)\uc774\uace0 \\(u_n\\to0\\)\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c \uad50\ub300\uae09\uc218 \\(\\sum_{n=1}^{\\infty}(-1)^n u_n\\)\uc758 \ud569\uc744 \\(S\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.
            \n\\[
            \n\\left|\\sum_{k=1}^n(-1)^k u_k-S\\right|\\le u_{n+1}.
            \n\\]<\/p>\n<\/div>\n

            \uadf8 \ubc16\uc758 \ud310\uc815\ubc95<\/h3>\n

            \ub2e4\uc591\ud55c \ubb34\ud55c\uae09\uc218\ub97c \ud310\uc815\ud558\ub294 \uace0\uae09 \ud310\uc815\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

            \n

            \uc815\ub9ac 7.11. (\ub77c\ube44 \ud310\uc815\ubc95)<\/span><\/p>\n

            \uc218\uc5f4 \\(\\{a_n\\}\\)\uc774 \\(a_n>0\\)\uc744 \ub9cc\uc871\ud558\uace0
            \n\\[
            \n\\lim_{n\\to\\infty}n\\left(1-\\frac{a_{n+1}}{a_n}\\right)=L\\in[-\\infty,\\infty]
            \n\\]
            \n\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

              \n
            1. \\(L>1\\)\uc774\uba74 \ubb34\ud55c\uae09\uc218 \\(\\sum a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/li>\n
            2. \\(L<1\\)\uc774\uba74 \ubb34\ud55c\uae09\uc218 \\(\\sum a_n\\)\uc774 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n
            3. \\(L=1\\)\uc774\uba74 \uc774 \ud310\uc815\ubc95\ub9cc\uc73c\ub85c\ub294 \\(\\sum a_n\\)\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \uacb0\uc815\ud560 \uc218 \uc5c6\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
              \n

              \uc99d\uba85<\/span>
              \n\uba3c\uc800 \\(L>1\\)\uc774\ub77c\uace0 \ud558\uc790. \\(1<s<r<L\\)\uc778 \uc2e4\uc218 \\(s,r\\)\uc744 \ud0dd\ud55c\ub2e4. \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0\uc11c
              \n\\[
              \n\\frac{a_{n+1}}{a_n}<1-\\frac rn.
              \n\\]
              \n\\(b_n=n^{-s}\\)\ub77c\uace0 \ud558\uc790. \ubb38\uc81c 5.19<\/a>\uc758 \uc2e4\uc218 \uc9c0\uc218 \ubca0\ub974\ub204\uc774 \ubd80\ub4f1\uc2dd\uc744 \\(x=-1\/(n+1)\\)\uc5d0 \uc801\uc6a9\ud558\uba74
              \n\\[
              \n\\frac{b_{n+1}}{b_n}
              \n=\\left(1-\\frac1{n+1}\\right)^s
              \n\\ge1-\\frac{s}{n+1}.
              \n\\]
              \n\ub610\ud55c \\(r>s\\)\uc774\ubbc0\ub85c \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0\uc11c
              \n\\[
              \n1-\\frac rn<1-\\frac{s}{n+1}\\le\\frac{b_{n+1}}{b_n}.
              \n\\]
              \n\ub530\ub77c\uc11c \\(a_n\/b_n\\)\uc740 \uacb0\uad6d \uac10\uc18c\ud558\ubbc0\ub85c \uc5b4\ub5a4 \\(C>0\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n\\le Cn^{-s}\\)\uc774\ub2e4. \\(s>1\\)\uc774\uace0 \uc815\ub9ac 7.7\uc5d0 \uc758\ud574 \\(\\sum a_n\\)\uc740 \uc218\ub834\ud55c\ub2e4.<\/p>\n

              \uc774\uc81c \\(L<1\\)\uc774\ub77c\uace0 \ud558\uc790. \\(L<r<s<1\\)\uc774\uba74\uc11c \\(0<r<s\\)\uc778 \\(r,s\\)\uc744 \ud0dd\ud560 \uc218 \uc788\ub2e4. \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0\uc11c
              \n\\[
              \n\\frac{a_{n+1}}{a_n}>1-\\frac rn.
              \n\\]
              \n\ub2e4\uc2dc \\(b_n=n^{-s}\\)\ub77c\uace0 \ud558\uba74 \\(0<s<1\\)\uc774\ubbc0\ub85c \uc2e4\uc218 \uc9c0\uc218 \ubca0\ub974\ub204\uc774 \ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud574
              \n\\[
              \n\\frac{b_{n+1}}{b_n}
              \n=\\left(1-\\frac1{n+1}\\right)^s
              \n\\le1-\\frac{s}{n+1}.
              \n\\]
              \n\\(s>r\\)\uc774\ubbc0\ub85c \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0\uc11c
              \n\\[
              \n1-\\frac rn>1-\\frac{s}{n+1}\\ge\\frac{b_{n+1}}{b_n}.
              \n\\]
              \n\ub530\ub77c\uc11c \\(a_n\/b_n\\)\uc740 \uacb0\uad6d \uc99d\uac00\ud558\uace0 \uc5b4\ub5a4 \\(C>0\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n\\ge Cn^{-s}\\)\uc774\ub2e4. \\(s<1\\)\uc774\ubbc0\ub85c \\(\\sum n^{-s}\\)\uac00 \ubc1c\uc0b0\ud558\uba70, \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud574 \\(\\sum a_n\\)\ub3c4 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

              \ub9c8\uc9c0\ub9c9\uc73c\ub85c \\(a_n=1\/n\\)\uacfc \\(a_n=1\/[n(\\ln n)^2]\\) (\\(n\\ge2\\))\uc5d0\uc11c\ub294 \ubaa8\ub450 \ub77c\ube44 \ud310\uc815\ubc95\uc758 \uadf9\ud55c\uc774 \\(1\\)\uc774\ub2e4. \uc55e \uae09\uc218\ub294 \ubc1c\uc0b0\ud558\uace0 \ub4a4 \uae09\uc218\ub294 \uc801\ubd84 \ud310\uc815\ubc95\uc73c\ub85c \uc218\ub834\ud558\ubbc0\ub85c \\(L=1\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \uacb0\ub860\uc744 \ub0b4\ub9b4 \uc218 \uc5c6\ub2e4.<\/span><\/p>\n<\/div>\n

              \ub2e4\uc74c \uc815\ub9ac\ub294 \ub450 \uc218\uc5f4\uc758 \uacf1\uc758 \ubb34\ud55c\uae09\uc218\ub97c \ud310\uc815\ud558\ub294 \uc720\uc6a9\ud55c \ubc29\ubc95\uc774\ub2e4.<\/p>\n

              \n

              \uc815\ub9ac 7.12. (\ub514\ub9ac\ud074\ub808 \ud310\uc815\ubc95)<\/span><\/p>\n

              \\(\\{a_n\\}\\)\uc774 \ub2e8\uc870\uc774\uace0 \\(a_n\\to0\\)\uc774\uba70, \\(\\sum b_n\\)\uc758 \ubd80\ubd84\ud569\uc774 \uc720\uacc4\uc774\uba74 \\(\\sum a_n b_n\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n

              \n

              \uc99d\uba85<\/span>
              \n\\(\\{a_n\\}\\)\uc774 \ub2e8\uc870\uac10\uc18c\ud558\ub294 \uacbd\uc6b0\ub97c \uba3c\uc800 \ubcf4\uc790. \uc774\ub54c \\(a_n\\to0\\)\uc774\ubbc0\ub85c \\(a_n\\ge0\\)\uc774\ub2e4. \\(B_n=\\sum_{k=1}^n b_k\\), \\(B_0=0\\)\uc774\ub77c\uace0 \ud558\uace0 \\(|B_n|\\le M\\)\uc778 \\(M>0\\)\uc744 \ud0dd\ud55c\ub2e4. \uadf8\ub7ec\uba74
              \n\\[
              \n\\begin{aligned}
              \n\\sum_{k=n}^m a_kb_k
              \n&=\\sum_{k=n}^m a_k(B_k-B_{k-1})\\\\
              \n&=a_mB_m-a_nB_{n-1}
              \n+\\sum_{k=n}^{m-1}(a_k-a_{k+1})B_k.
              \n\\end{aligned}
              \n\\]
              \n\uc704 \ub4f1\uc2dd\uc744 \uc544\ubca8\uc758 \ubd80\ubd84\ud569 \uacf5\uc2dd<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub530\ub77c\uc11c
              \n\\[
              \n\\begin{aligned}
              \n\\left|\\sum_{k=n}^m a_kb_k\\right|
              \n&\\le M\\left(a_m+a_n+\\sum_{k=n}^{m-1}(a_k-a_{k+1})\\right)\\\\
              \n&=M(a_m+a_n+a_n-a_m)=2Ma_n.
              \n\\end{aligned}
              \n\\]
              \n\\(a_n\\to0\\)\uc774\ubbc0\ub85c \uc815\ub9ac 7.2\uc5d0 \uc758\ud574 \\(\\sum a_nb_n\\)\uc740 \uc218\ub834\ud55c\ub2e4.<\/p>\n

              \\(\\{a_n\\}\\)\uc774 \ub2e8\uc870\uc99d\uac00\ud558\uc5ec \\(0\\)\uc73c\ub85c \uc218\ub834\ud558\ub294 \uacbd\uc6b0\uc5d0\ub294 \\(\\{-a_n\\}\\)\uc774 \ub2e8\uc870\uac10\uc18c\ud558\uc5ec \\(0\\)\uc73c\ub85c \uc218\ub834\ud558\ubbc0\ub85c \uc55e\uc758 \uacb0\uacfc\ub97c \\(-a_n\\)\uc5d0 \uc801\uc6a9\ud558\uba74 \ub41c\ub2e4.<\/span><\/p>\n<\/div>\n

              \ub450 \uc218\uc5f4\uc758 \uacf1\uc758 \ubb34\ud55c\uae09\uc218\ub97c \ud310\uc815\ud558\ub294 \ub610 \ub2e4\ub978 \ubc29\ubc95\uc774 \uc788\ub2e4.<\/p>\n

              \n

              \uc815\ub9ac 7.13. (\uc544\ubca8 \ud310\uc815\ubc95)<\/span><\/p>\n

              \\(\\{a_n\\}\\)\uc774 \ub2e8\uc870\uc774\uace0 \uc720\uacc4\uc774\uba70 \\(\\sum b_n\\)\uc774 \uc218\ub834\ud558\uba74 \\(\\sum a_n b_n\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n

              \n

              \uc99d\uba85<\/span>
              \n\\(\\{a_n\\}\\)\uc774 \ub2e8\uc870\uc720\uacc4\uc774\ubbc0\ub85c \uc5b4\ub5a4 \uc2e4\uc218 \\(a\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n\\to a\\)\uc774\ub2e4. \ub530\ub77c\uc11c
              \n\\[
              \n\\sum a_nb_n=a\\sum b_n+\\sum(a_n-a)b_n.
              \n\\]
              \n\uccab \ubc88\uc9f8 \uae09\uc218\ub294 \uc218\ub834\ud55c\ub2e4. \ub610\ud55c \\(\\{a_n-a\\}\\)\uc740 \ub2e8\uc870\uc774\uace0 \\(0\\)\uc73c\ub85c \uc218\ub834\ud558\uba70, \\(\\sum b_n\\)\uc774 \uc218\ub834\ud558\ubbc0\ub85c \uadf8 \ubd80\ubd84\ud569\uc740 \uc720\uacc4\uc774\ub2e4. \ub514\ub9ac\ud074\ub808 \ud310\uc815\ubc95\uc5d0 \uc758\ud574 \ub450 \ubc88\uc9f8 \uae09\uc218\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/span><\/p>\n<\/div>\n

              \ubb34\ud55c\uae09\uc218\uc758 \uc7ac\ubc30\uc5f4<\/h3>\n

              \ubb34\ud55c\uae09\uc218\uc758 \ud56d\ub4e4\uc758 \uc21c\uc11c\ub97c \ubc14\uafb8\ub294 \uac83\uc744 \uc7ac\ubc30\uc5f4<\/span>(rearrangement)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub354 \uc815\ud655\ud788 \ub9d0\ud558\uba74 \uc790\uc5f0\uc218 \uc804\uccb4\uc758 \uc9d1\ud569 \\(\\mathbb N\\)\uc5d0\uc11c \uc790\uc2e0\uc73c\ub85c\uc758 \uc77c\ub300\uc77c\ub300\uc751
              \n\\[
              \n\\sigma\\colon\\mathbb N\\to\\mathbb N
              \n\\]
              \n\uc5d0 \ub300\ud558\uc5ec
              \n\\[
              \n\\sum_{n=1}^{\\infty}a_{\\sigma(n)}
              \n\\]
              \n\uc744 \\(\\sum_{n=1}^{\\infty}a_n\\)\uc758 \uc7ac\ubc30\uc5f4 \uae09\uc218\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

              \uc720\ud55c\ud569\uc5d0\uc11c\ub294 \ud56d\uc758 \uc21c\uc11c\ub97c \ubc14\uafb8\uc5b4\ub3c4 \ud569\uc774 \ubcc0\ud558\uc9c0 \uc54a\uc9c0\ub9cc, \ubb34\ud55c\uae09\uc218\uc5d0\uc11c\ub294 \uc7ac\ubc30\uc5f4\uc774 \uc218\ub834\uc131\uacfc \ud569\uc5d0 \uc601\ud5a5\uc744 \uc904 \uc218 \uc788\ub2e4. \uc808\ub300\uc218\ub834\ud558\ub294 \uae09\uc218\uc5d0\uc11c\ub294 \uc774\ub7ec\ud55c \ubb38\uc81c\uac00 \uc0dd\uae30\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n

              \n

              \uc815\ub9ac 7.14. (\uc808\ub300\uc218\ub834 \uae09\uc218\uc758 \uc7ac\ubc30\uc5f4)<\/span><\/p>\n

              \ubb34\ud55c\uae09\uc218 \\(\\sum a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud558\uba74 \uc784\uc758\uc758 \uc7ac\ubc30\uc5f4 \uae09\uc218 \\(\\sum a_{\\sigma(n)}\\)\ub3c4 \uc808\ub300\uc218\ub834\ud558\uace0, \ubcf8\ub798\uc758 \uae09\uc218\uc640 \uac19\uc740 \ud569\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n

              \n

              \uc99d\uba85<\/span>
              \n\\(\\sum|a_n|\\)\uc758 \ud569\uc744 \\(A\\)\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(N\\)\uc5d0 \ub300\ud558\uc5ec
              \n\\[
              \n\\sum_{n=1}^N|a_{\\sigma(n)}|\\le A
              \n\\]
              \n\uc774\ubbc0\ub85c \uc7ac\ubc30\uc5f4\ub41c \uc808\ub313\uac12 \uae09\uc218\uc758 \ubd80\ubd84\ud569\uc740 \ub2e8\uc870\uc99d\uac00\ud558\uace0 \uc704\ub85c \uc720\uacc4\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\sum|a_{\\sigma(n)}|\\)\uc740 \uc218\ub834\ud55c\ub2e4.<\/p>\n

              \uc774\uc81c \\(S=\\sum a_n\\)\uc774\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec
              \n\\[
              \n\\sum_{n=N+1}^{\\infty}|a_n|<\\frac{\\varepsilon}{2}
              \n\\]
              \n\uc774 \ub418\ub3c4\ub85d \\(N\\)\uc744 \ud0dd\ud55c\ub2e4. \uadf8\ub9ac\uace0
              \n\\[
              \nM=\\max\\{\\sigma^{-1}(1),\\ldots,\\sigma^{-1}(N)\\}
              \n\\]
              \n\uc774\ub77c\uace0 \ud558\uc790. \\(m\\ge M\\)\uc774\uba74 \uc6d0\ub798 \uae09\uc218\uc758 \ucc98\uc74c \\(N\\)\uac1c \ud56d\uc774 \uc7ac\ubc30\uc5f4 \uae09\uc218\uc758 \ucc98\uc74c \\(m\\)\uac1c \ud56d \uc548\uc5d0 \ubaa8\ub450 \ub4e4\uc5b4 \uc788\uc73c\ubbc0\ub85c
              \n\\[
              \n\\left|\\sum_{n=1}^m a_{\\sigma(n)}-\\sum_{n=1}^N a_n\\right|
              \n\\le\\sum_{n=N+1}^{\\infty}|a_n|<\\frac{\\varepsilon}{2}.
              \n\\]
              \n\ub610\ud55c
              \n\\[
              \n\\left|S-\\sum_{n=1}^N a_n\\right|<\\frac{\\varepsilon}{2}.
              \n\\]
              \n\ub530\ub77c\uc11c \ucda9\ubd84\ud788 \ud070 \\(m\\)\uc5d0\uc11c
              \n\\[
              \n\\left|\\sum_{n=1}^m a_{\\sigma(n)}-S\\right|<\\varepsilon,
              \n\\]
              \n\uc989 \uc7ac\ubc30\uc5f4 \uae09\uc218\uc758 \ud569\ub3c4 \\(S\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

              \uc870\uac74\uc218\ub834\ud558\ub294 \uae09\uc218\uc758 \uacbd\uc6b0 \uc0c1\ud669\uc774 \ub2e4\ub974\ub2e4. \ub2e4\uc74c\uc740 \uae09\uc218\uc758 \uc7ac\ubc30\uc5f4\uc5d0 \uad00\ud55c \ub180\ub77c\uc6b4 \uacb0\uacfc\uc774\ub2e4.<\/p>\n

              \n

              \uc815\ub9ac 7.15. (\ub9ac\ub9cc \uc7ac\ubc30\uc5f4 \uc815\ub9ac)<\/span><\/p>\n

              \uc2e4\uc218\uae09\uc218 \\(\\sum a_n\\)\uc774 \uc870\uac74\uc218\ub834\ud558\uba74, \uc784\uc758\uc758 \uc2e4\uc218 \\(L\\)\uc5d0 \ub300\ud558\uc5ec \\(\\sum a_{\\sigma(n)}=L\\)\uc774 \ub418\ub3c4\ub85d \ud558\ub294 \uc7ac\ubc30\uc5f4 \\(\\sigma\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \ub610\ud55c \uae09\uc218\uac00 \\(+\\infty\\) \ub610\ub294 \\(-\\infty\\)\ub85c \ubc1c\uc0b0\ud558\ub3c4\ub85d \ud558\ub294 \uc7ac\ubc30\uc5f4\ub3c4 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n

              \n

              \uc99d\uba85<\/span>
              \n\\[
              \np_n=\\max\\{a_n,0\\},\\quad q_n=\\max\\{-a_n,0\\}
              \n\\]
              \n\uc774\ub77c\uace0 \ub450\uc790. \uadf8\ub7ec\uba74 \\(a_n=p_n-q_n\\), \\(|a_n|=p_n+q_n\\)\uc774\ub2e4. \ubd80\ubd84\ud569\uc744 \uac01\uac01 \\(P_N=\\sum_{n=1}^N p_n\\), \\(Q_N=\\sum_{n=1}^N q_n\\)\uc774\ub77c\uace0 \ud558\uba74
              \n\\[
              \nP_N-Q_N\\longrightarrow\\sum a_n,
              \n\\]
              \n\ud55c\ud3b8 \\(P_N+Q_N\\)\uc740 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ub2e4. \ub9cc\uc57d \\(P_N\\)\uacfc \\(Q_N\\) \uac00\uc6b4\ub370 \ud558\ub098\uac00 \uc704\ub85c \uc720\uacc4\ub77c\uba74 \uadf8\uac83\uc740 \ub2e8\uc870\uc99d\uac00\uc218\uc5f4\uc774\ubbc0\ub85c \uc218\ub834\ud558\uace0, \\(P_N-Q_N\\)\uc758 \uc218\ub834\uc131 \ub54c\ubb38\uc5d0 \ub2e4\ub978 \ud558\ub098\ub3c4 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\uba74 \\(P_N+Q_N\\)\ub3c4 \uc218\ub834\ud558\uc5ec \uc808\ub300\uc218\ub834\ud558\uac8c \ub418\ubbc0\ub85c \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c
              \n\\[
              \n\\sum p_n=\\sum q_n=\\infty.
              \n\\]<\/p>\n

              \uc591\uc758 \ud56d\ub4e4\uc744 \uc6d0\ub798 \uc21c\uc11c\ub300\ub85c \\(u_1,u_2,\\ldots\\), \uc74c\uc758 \ud56d\ub4e4\uc758 \uc808\ub313\uac12\uc744 \uc6d0\ub798 \uc21c\uc11c\ub300\ub85c \\(v_1,v_2,\\ldots\\)\ub77c\uace0 \ud558\uc790. \uc870\uac74\uc218\ub834\uae09\uc218\uc758 \uc77c\ubc18\ud56d\uc740 \\(0\\)\uc73c\ub85c \uc218\ub834\ud558\ubbc0\ub85c \\(u_j\\to0\\), \\(v_j\\to0\\)\uc774\ub2e4.<\/p>\n

              \uc2e4\uc218 \\(L\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \ud604\uc7ac \ubd80\ubd84\ud569\uc774 \\(L\\) \uc774\ud558\uc774\uba74 \uc544\uc9c1 \uc0ac\uc6a9\ud558\uc9c0 \uc54a\uc740 \uc591\uc758 \ud56d\uc744 \ucc28\ub840\ub85c \ub354\ud558\uc5ec \ucc98\uc74c\uc73c\ub85c \\(L\\)\uc744 \ucd08\uacfc\ud560 \ub54c\uae4c\uc9c0 \uc9c4\ud589\ud558\uace0, \ud604\uc7ac \ubd80\ubd84\ud569\uc774 \\(L\\) \uc774\uc0c1\uc774\uba74 \uc544\uc9c1 \uc0ac\uc6a9\ud558\uc9c0 \uc54a\uc740 \uc74c\uc758 \ud56d\uc744 \ucc28\ub840\ub85c \ub354\ud558\uc5ec \ucc98\uc74c\uc73c\ub85c \\(L\\)\ubcf4\ub2e4 \uc791\uc544\uc9c8 \ub54c\uae4c\uc9c0 \uc9c4\ud589\ud55c\ub2e4. \uc774 \ub450 \ub2e8\uacc4\ub97c \ubc18\ubcf5\ud55c\ub2e4. \\(\\sum u_j=\\sum v_j=\\infty\\)\uc774\ubbc0\ub85c \uac01 \ub2e8\uacc4\ub294 \uc720\ud55c \ubc88\uc758 \ub367\uc148 \ub4a4\uc5d0 \ub05d\ub098\uace0 \uacfc\uc815\uc744 \ubb34\ud55c\ud788 \uacc4\uc18d\ud560 \uc218 \uc788\ub2e4.<\/p>\n

              \uc591\uc758 \ud56d\uc744 \ub354\ud574 \\(L\\)\uc744 \ucc98\uc74c \ucd08\uacfc\ud55c \uc21c\uac04\uc758 \ucd08\uacfc\ub7c9\uc740 \ub9c8\uc9c0\ub9c9\uc5d0 \ub354\ud55c \uc591\uc758 \ud56d\ubcf4\ub2e4 \uc791\uace0, \uc74c\uc758 \ud56d\uc744 \ub354\ud574 \\(L\\) \uc544\ub798\ub85c \ucc98\uc74c \ub0b4\ub824\uac04 \uc21c\uac04\uc758 \ubd80\uc871\ub7c9\uc740 \ub9c8\uc9c0\ub9c9\uc5d0 \ub354\ud55c \\(v_j\\)\ubcf4\ub2e4 \uc791\ub2e4. \uc774 \ud56d\ub4e4\uc740 \\(0\\)\uc73c\ub85c \uc218\ub834\ud558\ubbc0\ub85c \uac01 \uc804\ud658\uc810\uc758 \ubd80\ubd84\ud569\uc740 \\(L\\)\ub85c \uc218\ub834\ud55c\ub2e4. \uac01 \ube14\ub85d \uc548\uc758 \ubd80\ubd84\ud569\uc740 \ub450 \uc804\ud658\uc810 \uc0ac\uc774\uc5d0\uc11c \ub2e8\uc870\ub86d\uac8c \uc6c0\uc9c1\uc774\ubbc0\ub85c \uc804\uccb4 \ubd80\ubd84\ud569\ub3c4 \\(L\\)\ub85c \uc218\ub834\ud55c\ub2e4. \uc591\uc758 \ud56d\uacfc \uc74c\uc758 \ud56d\uc758 \ucca8\uc790\ub294 \ub9e4 \ub2e8\uacc4 \uc55e\uc73c\ub85c \uc9c4\ud589\ud558\ubbc0\ub85c \\(0\\)\uc774 \uc544\ub2cc \ubaa8\ub4e0 \ud56d\uc774 \uc0ac\uc6a9\ub41c\ub2e4. \\(0\\)\uc778 \ud56d\uc774 \uc788\uc73c\uba74 \uc774 \uacfc\uc815 \uc0ac\uc774\uc5d0 \ucc28\ub840\ub85c \uc0bd\uc785\ud558\uba74 \ub41c\ub2e4. \ub530\ub77c\uc11c \uc774\ub294 \uc2e4\uc81c \uc7ac\ubc30\uc5f4\uc774\ub2e4.<\/p>\n

              \\(+\\infty\\)\ub85c \ubc1c\uc0b0\ud558\ub294 \uc7ac\ubc30\uc5f4\ub3c4 \ube44\uc2b7\ud558\uac8c \ub9cc\ub4e0\ub2e4. \\(k\\)\ubc88\uc9f8 \ub2e8\uacc4\uc5d0\uc11c \uc544\uc9c1 \uc0ac\uc6a9\ud558\uc9c0 \uc54a\uc740 \uc591\uc758 \ud56d\uc744 \uc801\uc5b4\ub3c4 \ud558\ub098 \uc0ac\uc6a9\ud558\uace0 \ubd80\ubd84\ud569\uc774 \\(k\\)\ub97c \ub118\uc744 \ub54c\uae4c\uc9c0 \ub354\ud55c \ub4a4, \uc544\uc9c1 \uc0ac\uc6a9\ud558\uc9c0 \uc54a\uc740 \uc74c\uc758 \ud56d \ud558\ub098\ub97c \ub354\ud55c\ub2e4. \\(v_k\\to0\\)\uc774\ubbc0\ub85c \ucda9\ubd84\ud788 \ud070 \\(k\\)\uc5d0\uc11c\ub294 \uc774 \uc74c\uc758 \ud56d\uc744 \ub354\ud55c \ub4a4\uc5d0\ub3c4 \ubd80\ubd84\ud569\uc774 \\(k-1\\)\ubcf4\ub2e4 \ud06c\ub2e4. \ub530\ub77c\uc11c \uc804\uccb4 \ubd80\ubd84\ud569\uc740 \\(+\\infty\\)\ub85c \uac04\ub2e4. \ub9e4 \ub2e8\uacc4 \uc74c\uc758 \ud56d\uc744 \ud558\ub098\uc529 \uc0ac\uc6a9\ud558\ubbc0\ub85c \ubaa8\ub4e0 \uc74c\uc758 \ud56d\ub3c4 \uacb0\uad6d \uc0ac\uc6a9\ub41c\ub2e4. \\(-\\infty\\)\uc758 \uacbd\uc6b0\uc5d0\ub294 \uc591\uc758 \ud56d\uacfc \uc74c\uc758 \ud56d\uc758 \uc5ed\ud560\uc744 \ubc14\uafb8\uba74 \ub41c\ub2e4.<\/span><\/p>\n<\/div>\n

              \n

              \ubb38\uc81c 7.6.<\/span>
              \n\uad50\ub300\uc870\ud654\uae09\uc218
              \n\\[
              \n\\sum_{n=1}^{\\infty}\\frac{(-1)^{n+1}}n
              \n\\]
              \n\uc744 \uc7ac\ubc30\uc5f4\ud558\uc5ec \uadf8 \ud569\uc774 \\(0\\)\uc774 \ub418\ub3c4\ub85d \ud558\ub294 \uc7ac\ubc30\uc5f4\uc744 \uad6c\uc131\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

              \n

              \ubb38\uc81c 7.7.<\/span>
              \n\ubb34\ud55c\uae09\uc218 \\(\\sum a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud560 \ub54c, \\(\\{a_n\\}\\)\uc758 \uc784\uc758\uc758 \uc7ac\ubc30\uc5f4\uc218\uc5f4 \\(\\{b_n\\}\\)\uc5d0 \ub300\ud558\uc5ec \\(\\sum a_nb_n\\)\uc774 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

              \n

              \ubb38\uc81c 7.8.<\/span>
              \n\ubb34\ud55c\uae09\uc218 \\(\\sum a_n\\)\uc774 \uc870\uac74\uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uc801\ub2f9\ud55c \uc7ac\ubc30\uc5f4 \\(\\sum a_{\\sigma(n)}\\)\uc744 \uad6c\uc131\ud558\uc5ec \uadf8 \ubd80\ubd84\ud569 \uc218\uc5f4\uc774 \\(+\\infty\\)\ub85c \uac00\ub294 \ubd80\ubd84\uc218\uc5f4\uacfc \\(-\\infty\\)\ub85c \uac00\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \ubaa8\ub450 \uac00\uc9c0\ub3c4\ub85d \ud558\uc2dc\uc624.<\/p>\n<\/div>\n

              \n

              \ubb38\uc81c 7.9.<\/span>
              \n\uc218\uc5f4 \\(\\{a_n\\}\\)\uc774 \\(a_1=1\\), \\(a_{n+1}=\\sin a_n\\)\uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c \ubb34\ud55c\uae09\uc218 \\(\\sum a_n\\)\uc774 \uc218\ub834\ud558\ub294\uc9c0 \ud310\ubcc4\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

              \n

              \ubb38\uc81c 7.10.<\/span>
              \n\uc790\uc5f0\uc0c1\uc218 \\(e\\)\ub97c \ubb34\ud55c\uae09\uc218\ub85c \ub098\ud0c0\ub0b4\ub824\uace0 \ud55c\ub2e4. \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n

                \n
              1. \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
                \n\\[
                \n\\left(1+\\frac1n\\right)^n\\le\\sum_{k=0}^n\\frac1{k!}
                \n\\]
                \n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n
              2. \\(m<n\\)\uc77c \ub54c
                \n\\[
                \n\\left(1+\\frac1n\\right)^n
                \n\\ge
                \n1+1+\\sum_{k=2}^m\\frac1{k!}
                \n\\left(1-\\frac1n\\right)
                \n\\left(1-\\frac2n\\right)\\cdots
                \n\\left(1-\\frac{k-1}{n}\\right)
                \n\\]
                \n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n
              3. \n\\[
                \ne=\\sum_{k=0}^{\\infty}\\frac1{k!}
                \n\\]
                \n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n
                \n

                \ubb38\uc81c 7.11.<\/span>
                \n\uc790\uc5f0\uc0c1\uc218 \\(e\\)\uac00 \ubb34\ub9ac\uc218\uc784\uc744 \ubcf4\uc774\ub824\uace0 \ud55c\ub2e4. \\(e=p\/q\\)\uc774\uace0 \\(p\\)\uc640 \\(q\\)\uac00 \uc11c\ub85c\uc18c\uc778 \uc790\uc5f0\uc218\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0
                \n\\[
                \nb_q=\\sum_{k=0}^q\\frac1{k!}
                \n\\]
                \n\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n

                  \n
                1. \ubd80\ub4f1\uc2dd
                  \n\\[
                  \n0<e-b_q
                  \n<\\frac1{(q+1)!}\\left(1+\\frac1{q+1}+\\frac1{(q+1)^2}+\\cdots\\right)
                  \n=\\frac1{q!q}
                  \n\\]
                  \n\uc774 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n
                2. \\(0<q!(e-b_q)<1\/q\\le1\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n
                3. \\(q!e\\)\uc640 \\(q!b_q\\)\uac00 \ubaa8\ub450 \uc815\uc218\uc784\uc744 \uc774\uc6a9\ud558\uc5ec \ubaa8\uc21c\uc744 \uc720\ub3c4\ud558\uace0, \\(e\\)\uac00 \ubb34\ub9ac\uc218\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n
                  \n

                  \ubb38\uc81c 7.12.<\/span>
                  \n\uc218\uc5f4 \\(\\{a_n\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \uc591\uc218\uc774\uace0 \\(a_n\\to0\\)\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n

                    \n
                  1. \ubb34\ud55c\uae09\uc218 \\(\\sum\\sin a_n\\)\uc774 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ubb34\ud55c\uae09\uc218 \\(\\sum a_n\\)\uc774 \uc218\ub834\ud558\ub294 \uac83\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n
                  2. \\(\\{a_n\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \uc591\uc218\ub77c\ub294 \uc870\uac74\uc744 \uc81c\uc678\ud574\ub3c4 (1)\uc774 \uc131\ub9bd\ud558\ub294\uc9c0 \ud310\ubcc4\ud558\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n
                    \n

                    \ud574\uc11d\ud559 \uac15\uc758\ub178\ud2b8<\/a><\/p>\n

                      \n
                    1. \uc2e4\uc218\uacc4\uc758 \uc131\uc9c8<\/a><\/li>\n
                    2. \uac70\ub9ac\uacf5\uac04<\/a><\/li>\n
                    3. \uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \uc704\uc0c1\uc801 \uc131\uc9c8<\/a><\/li>\n
                    4. \ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131<\/a><\/li>\n
                    5. \uc77c\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n
                    6. \uc77c\ubcc0\uc218 \ud568\uc218\uc758 \uc801\ubd84<\/a><\/li>\n
                    7. \ubb34\ud55c\uae09\uc218<\/a><\/li>\n
                    8. \uc2e4\ud574\uc11d\uc801 \ud568\uc218<\/a><\/li>\n
                    9. \ub2e4\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n
                    10. \uc911\uc801\ubd84<\/a><\/li>\n
                    11. \ubca1\ud130\uc7a5\uacfc \uc801\ubd84 \uc815\ub9ac<\/a><\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                      <\/p>\n","protected":false},"excerpt":{"rendered":"

                      \uc55e \uc7a5\ub4e4\uc5d0\uc11c\ub294 \uc218\uc5f4\uc758 \uc218\ub834\uacfc \ud568\uc218\uc758 \uadf9\ud55c, \ubbf8\ubd84, \uc801\ubd84\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc218\uc5f4\uc758 \ubd80\ubd84\ud569\uc744 \ub2e4\uc2dc \ud558\ub098\uc758 \uc218\uc5f4\ub85c \ubcf4\uc544 \ubb34\ud55c\uae09\uc218\ub97c \uc815\uc758\ud558\uace0, \uc218\uc5f4\uc758 \uc218\ub834 \uc774\ub860\uacfc \uc5ec\ub7ec \uc218\ub834 \ud310\uc815\ubc95\uc744 \uc804\uac1c\ud55c\ub2e4. \ud2b9\ubcc4\ud55c \uc5b8\uae09\uc774 \uc5c6\uc73c\uba74 \uc774 \uc7a5\uc758 \uc218\uc5f4\uc740 \uc2e4\uc218\uc5f4\ub85c \uac00\uc815\ud55c\ub2e4. \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834\uacfc \ubc1c\uc0b0 \uc218\uc5f4 \\(\\{a_n\\}\\)\uc5d0 \ub300\ud558\uc5ec \\( S_N=\\sum_{n=1}^N a_n \\) \uc744 \\(N\\)\ubc88\uc9f8 \ubd80\ubd84\ud569(partial sum)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uae30\ud638 \\( \\sum_{n=1}^{\\infty}a_n \\) \uc744 \\(\\{a_n\\}\\)\uc758 \ubb34\ud55c\uae09\uc218(infinite series)\ub77c\uace0 \ubd80\ub978\ub2e4. \ubd80\ubd84\ud569 \uc218\uc5f4 \\(\\{S_N\\}\\)\uc774 \uc218\ub834\ud558\uba74 \uc774 \ubb34\ud55c\uae09\uc218\uac00…<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9470,"menu_order":107,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9491","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9491","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=9491"}],"version-history":[{"count":11,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9491\/revisions"}],"predecessor-version":[{"id":10112,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9491\/revisions\/10112"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9470"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9491"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}