{"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":"2025-10-21T16:08:51","modified_gmt":"2025-10-21T07:08:51","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

\uc774 \uc7a5\uc5d0\uc11c\ub294 \ubb34\ud55c\uae09\uc218\uc758 \ub73b\uacfc \uc5ec\ub7ec \uac00\uc9c0 \ud310\uc815\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

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

\uc218\uc5f4 \\(\\{a_n\\}\\)\uc5d0 \ub300\ud558\uc5ec \ubd80\ubd84\ud569<\/span>
\n\\[S_N = \\sum_{n=1}^{N} a_n\\]
\n\uc5d0 \\(N\\rightarrow\\infty\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud588\uc744 \ub54c, \uadf8 \uadf9\ud55c\uc744 \\(\\left\\{a_n\\right\\}\\)\uc758 \ubb34\ud55c\uae09\uc218<\/span>(infinite series)\ub77c\uace0 \ubd80\ub974\uace0 \\(\\sum_{n=1}^{\\infty} a_n\\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989
\n\\[\\sum_{n=1}^{\\infty} a_n = \\lim_{N\\rightarrow\\infty} \\sum_{n=1}^{N} a_n \\]
\n\uc774\ub2e4. \uc774 \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\uba74 “\ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834<\/span>\ud55c\ub2e4(converge)”\ub77c\uace0 \ub9d0\ud558\uace0, \uadf8 \uadf9\ud55c\uac12\uc744 \uae09\uc218\uc758 \ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\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

\ubb34\ud55c\uae09\uc218\ub97c \uc815\uc758\ud560 \ub54c \uc218\uc5f4\uc758 \ud56d\ubc88\ud638\uac00 \ubc18\ub4dc\uc2dc \\(1\\)\ubd80\ud130 \uc2dc\uc791\ud558\ub294 \uac83\uc740 \uc544\ub2c8\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \ub2e4\uc74c\uc740 \ubaa8\ub450 \ubb34\ud55c\uae09\uc218\uc774\ub2e4.
\n\\[\\sum_{n=2}^{\\infty} a_n ,\\quad \\sum_{k=0}^{\\infty} b_k ,\\quad \\sum_{j=4}^{\\infty} c_j .\\]<\/p>\n

\\(\\left\\{a_n \\right\\}\\)\uc758 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud558\uace0 \uadf8 \ud569\uc774 \\(S\\)\uc774\uba74
\n\\[\\lim_{n\\rightarrow\\infty} a_n = \\lim_{n\\rightarrow\\infty} \\left\\{ \\left( \\sum_{k=1}^{n} a_k \\right) – \\left( \\sum_{k=1}^{n-1} a_k\\right)\\right\\} = \\sum_{k=1}^\\infty a_k – \\sum_{k=1}^\\infty a_k = S-S = 0\\]
\n\uc774\ubbc0\ub85c \\(a_n \\to 0\\)\uc774\ub2e4. \uadf8\ub7ec\ub098 \uc77c\ubc18\uc801\uc73c\ub85c \uc5ed\uc740 \ucc38\uc774 \uc544\ub2c8\ub2e4. \uc608\ub97c \ub4e4\uba74, \\(\\frac{1}{n}\\rightarrow 0\\)\uc774\uc9c0\ub9cc \uc870\ud654\uae09\uc218 \\(\\sum_{n=1}^{\\infty} \\frac{1}{n}\\)\uc740 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

\uc2e4\uc218\uc5f4\uc774 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc774 \ucf54\uc2dc \uc218\uc5f4\uc778 \uac83\ucc98\ub7fc \ubb34\ud55c\uae09\uc218\uc5d0\uc11c\ub3c4 \ucf54\uc2dc \uc870\uac74\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \uc989 \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 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(m > n > N\\)\uc77c \ub54c \\(\\left\\lvert\\sum_{k=n+1}^{m} a_k\\right\\rvert < \\varepsilon\\)\uc778 \uac83\uc774\ub2e4. \uc774\uc640 \uac19\uc740 \uc870\uac74\uc744 \ubb34\ud55c\uae09\uc218\uc5d0 \ub300\ud55c \ucf54\uc2dc \ud310\uc815\ubc95<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\ubb34\ud55c\uae09\uc218 \\(\\sum |a_n|\\)\uc774 \uc218\ub834\ud560 \ub54c “\ubb34\ud55c\uae09\uc218 \\(\\sum a_n\\)\uc774 \uc808\ub300\uc218\ub834<\/span>\ud55c\ub2e4(converges absolutely)”\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ub610\ud55c \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud558\uc9c0\ub9cc \uc808\ub300\uc218\ub834\ud558\uc9c0 \uc54a\uc744 \ub54c, “\ubb34\ud55c\uae09\uc218\uac00 \uc870\uac74\uc218\ub834<\/span>\ud55c\ub2e4(converges conditionally)”\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n

\uc808\ub300\uc218\ub834\ud558\ub294 \uae09\uc218\ub294 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ub098 \uadf8 \uc5ed\uc740 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \uad50\ub300\uc870\ud654\uae09\uc218 \\(\\sum (-1)^{n+1} \\frac{1}{n}\\)\uc740 \uc218\ub834\ud558\uc9c0\ub9cc \uc808\ub300\uc218\ub834\ud558\uc9c0\ub294 \uc54a\ub294\ub2e4.<\/p>\n

\n

\ubb38\uc81c 7.1.<\/span>
\n\ubb34\ud55c\uae09\uc218 \\(\\sum \\left\\lvert a_n \\right\\rvert\\)\uc774 \uc218\ub834\ud558\uba74 \ubb34\ud55c\uae09\uc218 \\(\\sum a_n\\)\ub3c4 \uc218\ub834\ud568\uc744 \uc99d\uba85\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 \\ge 0\\)\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 \\(\\sum_{k=1}^n a_k\\)\ub294 \ub2e8\uc870\uc99d\uac00\ud558\ub294 \uc218\uc5f4\uc774 \ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uc591\ud56d\uae09\uc218\uac00 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ubd80\ubd84\ud569 \uc218\uc5f4\uc774 \uc720\uacc4\uc778 \uac83\uc774\ub2e4.<\/p>\n

\uc774\ub7ec\ud55c \uc131\uc9c8\uc744 \uc0ac\uc6a9\ud558\uc5ec \ub450 \uc591\ud56d\uae09\uc218\ub97c \ube44\uad50\ud558\ub294 \ud310\uc815\ubc95\uc744 \uc720\ub3c4\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\n

\uc815\ub9ac 7.1. (\uc591\ud56d\uae09\uc218\uc758 \ube44\uad50 \ud310\uc815\ubc95)<\/span><\/p>\n

\uc784\uc758\uc758 \\(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<\/p>\n

\\(\\sum a_n\\)\uacfc \\(\\sum b_n\\)\uc758 \ubd80\ubd84\ud569\uc744 \uac01\uac01 \\(A_n\\), \\(B_n\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\left\\{ A_n \\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud558\ub294 \uc218\uc5f4\uc774\uace0 \\(a_n\\le b_n\\)\uc774\ubbc0\ub85c \\(A_n \\le B_n\\)\uc774\ub2e4. \\(\\left\\{ B_n \\right\\}\\)\uc774 \uc218\ub834\ud558\ubbc0\ub85c \\(\\left\\{ B_n \\right\\}\\)\uc740 \uc720\uacc4\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\left\\{ A_n \\right\\}\\)\ub3c4 \uc720\uacc4\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

\ube44\uc2b7\ud55c \ubc29\ubc95\uc73c\ub85c, \ub450 \uc591\ud56d\uae09\uc218\uc758 \ud56d\uc758 \ube44\uc758 \uadf9\ud55c\uc744 \uc0ac\uc6a9\ud558\ub294 \ubc29\ubc95\uc774 \uc788\ub2e4.<\/p>\n

\n

\uc815\ub9ac 7.2. (\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 \\ge 0\\), \\(b_n > 0\\)\uc774\uace0, \\(\\lim_{n \\to \\infty} \\frac{a_n}{b_n} = L\\)\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\uac00 \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<\/p>\n

    \\(0 < L < \\infty\\)\uc778 \uacbd\uc6b0\ub97c \uc99d\uba85\ud558\uc790. \\(\\varepsilon = L\/2\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n>N\\)\uc77c \ub54c
    \n\\[\\frac{1}{2}L = L-\\varepsilon < \\frac{a_n}{b_n} < L+\\varepsilon = \\frac{3}{2}L\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc774 \ubd80\ub4f1\uc2dd\uc744 \ubcc0\ud615\ud558\uba74\n\\[Lb_n <2a_n < 3Lb_n\\]\n\uc774\ubbc0\ub85c \ube44\uad50 \ud310\uc815\ubc95(\uc815\ub9ac 7.1)\uc5d0 \uc758\ud558\uc5ec \ubc14\ub77c\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

    \ub450 \uc218\uc5f4\uc744 \ube44\uad50\ud558\ub294 \uac83\uc774 \uc544\ub2c8\ub77c \uc218\uc5f4\uacfc \ud568\uc218\ub97c \ube44\uad50\ud558\ub294 \ubc29\ubc95\uc774 \uc788\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 7.3. (\uc801\ubd84 \ud310\uc815\ubc95)<\/span><\/p>\n

    \ud568\uc218 \\(f\\)\uac00 \\([1,\\, \\infty)\\)\uc5d0\uc11c \uac10\uc18c\ud558\uace0, \uc774 \uad6c\uac04\uc5d0\uc11c \\(f(x) \\ge 0\\)\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<\/p>\n

    \\(b_n = \\int_{n}^{n+1} f(x)dx\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(0\\le b_{n+1} \\le a_{n+1} \\le b_n\\)\uc774\ubbc0\ub85c \ube44\uad50 \ud310\uc815\ubc95(\uc815\ub9ac 7.1)\uc5d0 \uc758\ud558\uc5ec \ubc14\ub77c\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

    \uc801\ubd84 \ud310\uc815\ubc95\uc744 \uc0ac\uc6a9\ud558\uba74 \\(\\sum \\frac{1}{n^p}\\)\uc744 \ud310\uc815\ud560 \uc218 \uc788\ub2e4. \uc774 \ubb34\ud55c\uae09\uc218\ub294 \\(p>1\\)\uc77c \ub54c \uc218\ub834\ud558\uace0 \\(p\\le 1\\)\uc77c \ub54c \ubc1c\uc0b0\ud55c\ub2e4. \uc774\uc640 \uac19\uc740 \ubb34\ud55c\uae09\uc218 \\(\\sum \\frac{1}{n^p}\\)\uc744 \\(p\\)-\uae09\uc218\ub77c\uace0 \ubd80\ub974\uace0, \uc774 \ud310\uc815\ubc95\uc744 \\(p\\)-\uae09\uc218 \ud310\uc815\ubc95<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

    \n

    \ubb38\uc81c 7.2.<\/span>
    \n\\(\\left\\{ a_n \\right\\}\\)\uc774 \uac10\uc18c\ud558\ub294 \uc218\uc5f4\uc774\uace0 \\(a_n \\ge 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub450 \ubb34\ud55c\uae09\uc218 \\(\\sum a_n\\)\uacfc \\(\\sum 2^n a_{2^n}\\)\uc740 \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. \\(\\sum_{n=1}^{\\infty} \\frac{1}{n}\\)<\/li>\n
    2. \\(\\sum_{n=1}^{\\infty} \\frac{1}{\\sqrt{n}}\\)<\/li>\n
    3. \\(\\sum_{n=1}^{\\infty} \\frac{1}{n^2}\\)<\/li>\n
    4. \\(\\sum_{n=1}^{\\infty} \\frac{n}{3^n}\\)<\/li>\n
    5. \\(\\sum_{n=1}^{\\infty} \\frac{n+1}{n^3 -n+1}\\)<\/li>\n
    6. \\(\\sum_{n=1}^{\\infty} \\frac{n^2 +n}{n^3 -2n+2}\\)<\/li>\n
    7. \\(\\sum_{n=2}^{\\infty} \\frac{1}{n\\ln n}\\)<\/li>\n
    8. \\(\\sum_{n=2}^{\\infty} \\frac{1}{n(\\ln n)^2}\\)<\/li>\n
    9. \\(\\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 7.4. (\ube44 \ud310\uc815\ubc95)<\/span><\/p>\n

      \uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc774 \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(a_n \\ne 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\lim_{n \\to \\infty} \\left\\lvert\\frac{a_{n+1}}{a_n}\\right\\rvert = L\\)\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\uc73c\ub85c \\(\\sum a_n\\)\uc744 \ud310\uc815\ud560 \uc218 \uc5c6\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
        \n

        \uc99d\uba85<\/p>\n

        \\(L<1\\)\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \\(\\varepsilon = (1-L)\/2\\), \\(r=L+\\varepsilon\\)\uc774\ub77c\uace0 \ud558\uba74, \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n\\ge N\\)\uc77c \ub54c\n\\[\\left\\lvert \\frac{a_{n+1}}{a_n} \\right\\rvert < L+\\varepsilon =r < 1\\]\n\uc989 \\(\\lvert a_{n+1} \\rvert < r\\lvert a_n \\rvert\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc790\uc5f0\uc218 \\(k\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\\lvert a_{N+k}\\rvert < r^k \\lvert a_N \\rvert\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc774\ub54c \\(\\sum_k r^k \\lvert a_N \\rvert\\) \uc774 \uc218\ub834\ud558\ubbc0\ub85c \ube44\uad50 \ud310\uc815\ubc95(\uc815\ub9ac 7.1)\uc5d0 \uc758\ud558\uc5ec \\(\\sum_k \\lvert a_{N+k}\\rvert\\)\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/p>\n

        \ub2e4\uc74c\uc73c\ub85c \\(L>1\\) \ub610\ub294 \\(L=\\infty\\)\uc778 \uacbd\uc6b0, \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n>N\\)\uc77c \ub54c \\(\\lvert a_{n+1} \\rvert \\ge \\lvert a_n \\rvert\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\left\\{a_n \\right\\}\\)\uc774 \\(0\\)\uc73c\ub85c \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

        \uc774\uc640 \ube44\uc2b7\ud558\uac8c, \uc81c\uacf1\uadfc\uc744 \uc0ac\uc6a9\ud558\ub294 \ubc29\ubc95\uc774 \uc788\ub2e4.<\/p>\n

        \n

        \uc815\ub9ac 7.5. (\uc81c\uacf1\uadfc \ud310\uc815\ubc95)<\/span><\/p>\n

        \uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \\(\\displaystyle\\varlimsup_{n \\to \\infty} \\sqrt[n]{\\left\\lvert a_n \\right\\rvert} = L\\)\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\uc73c\ub85c \\(\\sum a_n\\)\uc744 \ud310\uc815\ud560 \uc218 \uc5c6\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
          \n

          \uc99d\uba85<\/p>\n

          \\(L<1\\)\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \\(\\varepsilon = (1-L)\/2\\), \\(r=L+\\varepsilon\\)\uc774\ub77c\uace0 \ud558\uba74, \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n\\ge N\\)\uc77c \ub54c\n\\[\\sqrt[n]{\\left\\lvert a_n \\right\\rvert} < L+\\varepsilon = r < 1\\]\n\uc989 \\(\\left\\lvert a_n \\right\\rvert < r^n\\)\uc774\ub2e4. \uc5ec\uae30\uc11c \\(\\sum_n r^n\\)\uc774 \uc218\ub834\ud558\ubbc0\ub85c \ube44\uad50 \ud310\uc815\ubc95(\uc815\ub9ac 7.1)\uc5d0 \uc758\ud558\uc5ec \\(\\sum_n \\lvert a_n \\rvert\\)\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/p>\n

          \ub2e4\uc74c\uc73c\ub85c \\(L>1\\) \ub610\ub294 \\(L=\\infty\\)\uc778 \uacbd\uc6b0, \\(\\sqrt[n]{\\lvert a_n \\rvert} > 1\\)\uc778 \ud56d \\(a_n\\)\uc774 \ubb34\ud55c\ud788 \ub9ce\uc73c\ubbc0\ub85c \\(\\left\\{ a_n \\right\\}\\)\uc740 \\(0\\)\uc73c\ub85c \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

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

          \uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc778 \uc218\uc5f4 \\(\\left\\{ u_n \\right\\}\\)\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, \\(\\left\\{ a_n \\right\\}\\)\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 7.6. (\uad50\ub300\uae09\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n

          \uc218\uc5f4 \\(\\left\\{u_n \\right\\}\\)\uc774 \uac10\uc18c\ud558\ub294 \uc591\ud56d\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uc989 \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(u_n \\ge u_{n+1} \\ge 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ubb34\ud55c\uae09\uc218 \\(\\sum_{n=1}^\\infty (-1)^n u_n \\)\uc774 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(u_n \\rightarrow 0\\)\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

          \n

          \uc99d\uba85<\/p>\n

          \\(\\sum (-1)^n u_n\\)\uc774 \uc218\ub834\ud558\ub294 \uacbd\uc6b0 \\(u_n \\rightarrow 0\\)\uc784\uc740 \uc774\ubbf8 \uc99d\uba85\ud558\uc600\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc5ed\uc744 \uc99d\uba85\ud558\uc790.<\/p>\n

          \\(u_n \\rightarrow 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(s_n = \\sum_{k=1}^{n} (-1)^k u_k\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\left\\{ s_{2n-1}\\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud558\uace0 \\(\\left\\{ s_{2n}\\right\\}\\)\uc740 \ub2e8\uc870\uac10\uc18c\ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(s_{2n-1} \\le s_{2n}\\)\uc774\ubbc0\ub85c, \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{ s_{2n-1}\\right\\}\\)\uacfc \\(\\left\\{ s_{2n}\\right\\}\\)\uc774 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4. \\(u_n \\rightarrow 0\\)\uc774\ubbc0\ub85c \\(\\left\\{ s_{2n-1}\\right\\}\\)\uacfc \\(\\left\\{ s_{2n}\\right\\}\\)\uc758 \uadf9\ud55c\uc774 \ub3d9\uc77c\ud558\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. \\(\\sum_{n=1}^{\\infty} \\frac{1}{n}x^n\\)<\/li>\n
          2. \\(\\sum_{n=1}^{\\infty} \\frac{n}{n+1}x^n\\)<\/li>\n
          3. \\(\\sum_{n=1}^{\\infty} \\frac{1}{n(n+1)}x^n\\)<\/li>\n
          4. \\(\\sum_{n=1}^{\\infty} \\frac{1}{n5^n}x^n\\)<\/li>\n
          5. \\(\\sum_{n=1}^{\\infty} \\frac{1}{n(n+1)(n+2)}x^{2n}\\)<\/li>\n
          6. \\(\\sum_{n=1}^{\\infty} \\frac{1}{(\\ln(n+1))^2}x^{n+1}\\)<\/li>\n
          7. \\(\\sum_{n=1}^{\\infty} \\frac{1}{1+n^3}x^n\\)<\/li>\n
          8. \\(\\sum_{n=1}^{\\infty} \\frac{1}{n^2}(x-4)^n\\)<\/li>\n
          9. \\(\\sum_{n=1}^{\\infty} \\frac{1}{5^n}(3x-2)^n\\)<\/li>\n<\/ol>\n<\/div>\n
            \n

            \ubb38\uc81c 7.5.<\/span>
            \n\uc218\uc5f4 \\(\\left\\{u_n \\right\\}\\)\uc774 \uac10\uc18c\ud558\uace0 \\(0\\)\uc5d0 \uc218\ub834\ud558\ub294 \uc218\uc5f4\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\\[\\left\\lvert \\,\\sum_{k=1}^n (-1)^k u_k – S \\,\\right\\rvert \\le \\left\\lvert u_{n+1} \\right\\rvert. \\]<\/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 7.7. (\ub77c\ube44 \ud310\uc815\ubc95)<\/span><\/p>\n

            \uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec, \\(a_n > 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0
            \n\\[\\lim_{n \\to \\infty} n\\left(1 – \\frac{a_{n+1}}{a_n}\\right) = L\\]
            \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\uc73c\ub85c \\(\\sum a_n\\)\uc744 \ud310\uc815\ud560 \uc218 \uc5c6\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
              \n

              \uc99d\uba85<\/p>\n

              \\(L > 1\\)\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \\(L > r > 1\\)\uc778 \\(r\\)\uc744 \uc120\ud0dd\ud558\uc790. \uadf8\ub7ec\uba74 \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud574
              \n\\[n\\left(1 – \\frac{a_{n+1}}{a_n}\\right) > r\\]
              \n\uc774\ubbc0\ub85c \\(\\frac{a_{n+1}}{a_n} < 1 - \\frac{r}{n}\\)\uc774\ub2e4. \\(n = N,\\, N+1,\\, \\ldots,\\, m-1\\)\uc77c \ub54c, \uc774 \ubd80\ub4f1\uc2dd\uc744 \ubcc0\ub9c8\ub2e4 \uacf1\ud558\uba74\n\\[\\frac{a_m}{a_N} < \\prod_{n=N}^{m-1} \\left(1 - \\frac{r}{n}\\right)\\]\n\uc744 \uc5bb\ub294\ub2e4. \uc6b0\ubcc0\uc758 \uacf1\uc744 \uc815\ub9ac\ud558\uba74\n\\[\\prod_{n=N}^{m-1} \\left(1 - \\frac{r}{n}\\right) = \\prod_{n=N}^{m-1} \\frac{n-r}{n} = \\frac{\\Gamma(N)\\Gamma(m-r)}{\\Gamma(m)\\Gamma(N-r)}\\]\n\uc774\ub2e4. \uc5ec\uae30\uc11c \uac10\ub9c8\ud568\uc218\uc758 \uc131\uc9c8(\uc2a4\ud0c8\ub9c1 \uadfc\uc0ac)\uc744 \uc0ac\uc6a9\ud558\uba74 \uc774 \uac12\uc774 \ub300\ub7b5 \\(Cm^{-r}\\) \ud615\ud0dc\uc784\uc744 \uc54c \uc218 \uc788\ub2e4.<\/p>\n

              \ub530\ub77c\uc11c \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud574 \\(|a_n| \\leq \\frac{C}{n^r}\\)\uc774\uace0, \\(r > 1\\)\uc774\ubbc0\ub85c \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud574 \\(\\sum a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n

              \ub2e4\uc74c\uc73c\ub85c \\(L < 1\\)\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \\(L < r < 1\\)\uc778 \\(r\\)\uc744 \uc120\ud0dd\ud558\uc790. \uadf8\ub7ec\uba74 \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud574\n\\[n\\left(1 - \\frac{a_{n+1}}{a_n}\\right) < r\\]\n\uc774\ubbc0\ub85c \\(\\frac{a_{n+1}}{a_n} > 1 – \\frac{r}{n}\\)\uc774\ub2e4.
              \n\uc55e\uc5d0\uc11c\uc640 \uc720\uc0ac\ud55c \ubc29\ubc95\uc73c\ub85c \\(|a_n| \\geq \\frac{C}{n^r}\\)\ub97c \uc5bb\ub294\ub2e4. \\(r < 1\\)\uc774\ubbc0\ub85c \\(\\sum \\frac{1}{n^r}\\)\uc774 \ubc1c\uc0b0\ud558\uace0, \ub530\ub77c\uc11c \\(\\sum a_n\\)\ub3c4 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

              \\(L = 1\\)\uc77c \ub54c\ub294 \\(\\sum \\frac{1}{n}\\)\uacfc \\(\\sum \\frac{1}{n^2}\\) \ubaa8\ub450 \uc774 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\uc9c0\ub9cc \ud558\ub098\ub294 \ubc1c\uc0b0\ud558\uace0 \ud558\ub098\ub294 \uc218\ub834\ud558\ubbc0\ub85c, \uc774 \ubc29\ubc95\uc73c\ub85c \ud310\uc815\uc774 \ubd88\uac00\ub2a5\ud558\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 7.8. (\ub514\ub9ac\ud074\ub808 \ud310\uc815\ubc95)<\/span><\/p>\n

              \\(\\{a_n\\}\\)\uc774 \ub2e8\uc870\uc774\uace0 \\(a_n \\to 0\\)\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<\/p>\n

              \\(\\left\\{ a_n \\right\\}\\)\uc774 \ub2e8\uc870\uac10\uc18c\ud558\ub294 \uacbd\uc6b0\ub9cc \uc0b4\ud3b4\ubd10\ub3c4 \ucda9\ubd84\ud558\ub2e4.<\/p>\n

              \\(\\sum b_n\\)\uc758 \ubd80\ubd84\ud569\uc744 \\(B_n = \\sum_{k=1}^n b_k\\)\ub77c\uace0 \ud558\uc790. \\(\\sum b_n\\)\uc758 \ubd80\ubd84\ud569\uc774 \uc720\uacc4\uc774\ubbc0\ub85c \\(M > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \ubaa8\ub4e0 \\(n\\)\uc5d0\uc11c \\(|B_n| \\leq M\\)\uc774\ub2e4. \\(B_0 = 0\\)\uc73c\ub85c \uc815\uc758\ud558\uba74
              \n\\[\\sum_{k=n}^m a_k b_k = \\sum_{k=n}^m a_k (B_k – B_{k-1}) = a_m B_m – a_{n-1} B_{n-1} + \\sum_{k=n}^{m-1} B_k(a_k – a_{k+1})\\]
              \n\uc744 \uc5bb\ub294\ub2e4. (\uc774 \ub4f1\uc2dd\uc744 \uc544\ubca8\uc758 \ubd80\ubd84\ud569 \uacf5\uc2dd<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.)<\/p>\n

              \\(\\{a_n\\}\\)\uc774 \ub2e8\uc870\uac10\uc18c\ud558\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
              \n\\[\\left|\\sum_{k=n}^m a_k b_k\\right| \\leq |a_m||B_m| + |a_{n-1}||B_{n-1}| + \\sum_{k=n}^{m-1} |B_k|(a_k – a_{k+1}).\\]
              \n\\(|B_k| \\leq M\\)\uc774\uace0 \\(a_k – a_{k+1} \\geq 0\\)\uc774\ubbc0\ub85c
              \n\\[\\left|\\sum_{k=n}^m a_k b_k\\right| \\leq M\\left(|a_m| + |a_{n-1}| + \\sum_{k=n}^{m-1} (a_k – a_{k+1})\\right)\\]
              \n\uc774\ub2e4. \uc5ec\uae30\uc11c
              \n\\[\\sum_{k=n}^{m-1} (a_k – a_{k+1}) = a_n – a_m\\]
              \n\uc774\ubbc0\ub85c \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
              \n\\[\\left|\\sum_{k=n}^m a_k b_k\\right| \\leq M(|a_m| + |a_{n-1}| + |a_n| + |a_m|) \\leq 2M(|a_{n-1}| + |a_m|).\\]
              \n\\(a_n \\to 0\\)\uc774\ubbc0\ub85c, \uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud574 \ucda9\ubd84\ud788 \ud070 \\(N\\)\uc744 \uc120\ud0dd\ud558\uba74 \\(m > n > N\\)\uc77c \ub54c
              \n\\[\\left|\\sum_{k=n}^m a_k b_k\\right| < \\varepsilon\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ucf54\uc2dc \ud310\uc815\ubc95\uc5d0 \uc758\ud574 \\(\\sum a_n b_n\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/span><\/p>\n<\/div>\n

              \ub450 \uc218\uc5f4\uc758 \uacf1\uc758 \ubb34\ud55c\uae09\uc218\ub97c \ud310\uc815\ud558\ub294 \ub610 \ub2e4\ub978 \uc720\uc6a9\ud55c \ubc29\ubc95\uc774 \uc788\ub2e4.<\/p>\n

              \n

              \uc815\ub9ac 7.9. (\uc544\ubca8 \ud310\uc815\ubc95)<\/span><\/p>\n

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

              \n

              \uc99d\uba85<\/p>\n

              \\(\\sum b_n\\)\uc774 \uc218\ub834\ud558\uba74 \ubd80\ubd84\ud569 \\(B_n\\)\uc774 \uc720\uacc4\uc774\uace0, \\(\\{a_n\\}\\)\uc774 \ub2e8\uc870\uc720\uacc4\uc774\uba74 \uc218\ub834\ud558\ubbc0\ub85c \\(a_n \\to a\\)\uc778 \\(a\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7ec\uba74 \\(\\sum a_n b_n = a\\sum b_n + \\sum (a_n – a)b_n\\)\uc73c\ub85c \uc4f8 \uc218 \uc788\uace0, \uc6b0\ubcc0\uc758 \ub450 \ubc88\uc9f8 \ubb34\ud55c\uae09\uc218\ub294 \ub514\ub9ac\ud074\ub808 \ud310\uc815\ubc95\uc758 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a8\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 \\(\\sigma : \\mathbb{N} \\to \\mathbb{N}\\)\uc5d0 \ub300\ud558\uc5ec
              \n\\[\\sum_{n=1}^{\\infty} a_{\\sigma(n)}\\]
              \n\uc744 \ubb34\ud55c\uae09\uc218 \\(\\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 \uc77c\ubc18\uc801\uc73c\ub85c \uc7ac\ubc30\uc5f4\uc774 \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834\uc131\uacfc \ud569\uc5d0 \uc601\ud5a5\uc744 \uc904 \uc218 \uc788\ub2e4. \uadf8\ub7ec\ub098 \uc808\ub300\uc218\ub834\ud558\ub294 \uae09\uc218\ub294 \uc7ac\ubc30\uc5f4\ud558\ub354\ub77c\ub3c4 \uc218\ub834\uc131\uacfc \ud569\uc774 \ubcc0\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n

              \n

              \uc815\ub9ac 7.10. (\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 \ubb34\ud55c\uae09\uc218\uc640 \uac19\uc740 \ud569\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n

              \n

              \uc99d\uba85<\/p>\n

              \\(\\sum a_n\\)\uc774 \uc808\ub300\uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uc989 \\(\\sum |a_n|\\)\uc774 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790.<\/p>\n

              \uba3c\uc800 \\(\\sum |a_{\\sigma(n)}|\\)\uc774 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uc790. \\(\\sum_{n=1}^N |a_{\\sigma(n)}|\\)\uc758 \ubd80\ubd84\ud569\uc744 \uc0dd\uac01\ud558\uba74, \\(\\sigma\\)\uac00 \uc77c\ub300\uc77c\ub300\uc751\uc774\ubbc0\ub85c \uc774 \ubd80\ubd84\ud569\uc740 \uc6d0\ub798 \uae09\uc218\uc758 \uc5b4\ub5a4 \uc720\ud55c\uac1c \ud56d\ub4e4\uc758 \uc808\ub313\uac12\uc758 \ud569\ubcf4\ub2e4 \uc791\uac70\ub098 \uac19\ub2e4. \ub530\ub77c\uc11c \ubd80\ubd84\ud569\uc774 \uc720\uacc4\uc774\uace0, \uc591\ud56d\uae09\uc218\uc774\ubbc0\ub85c \uc218\ub834\ud55c\ub2e4.<\/p>\n

              \ub2e4\uc74c\uc73c\ub85c \uc7ac\ubc30\uc5f4 \uae09\uc218\uc758 \ud569\uc774 \uc774\uc804 \uae09\uc218\uc640 \uc77c\uce58\ud568\uc744 \ubcf4\uc774\uc790. \\(S = \\sum a_n\\), \\(S’ = \\sum a_{\\sigma(n)}\\)\uc774\ub77c\uace0 \ud558\uc790. \uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec, \uc808\ub300\uc218\ub834\uc131\uc5d0 \uc758\ud574 \ucda9\ubd84\ud788 \ud070 \\(N\\)\uc744 \uc120\ud0dd\ud558\uba74
              \n\\[\\sum_{n=N+1}^{\\infty} |a_n| < \\varepsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \n\\(M = \\max\\{\\sigma^{-1}(1),\\, \\sigma^{-1}(2),\\, \\ldots,\\, \\sigma^{-1}(N)\\}\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(m > M\\)\uc77c \ub54c, \\(\\{a_1,\\, a_2,\\, \\ldots,\\, a_N\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(\\{a_{\\sigma(1)},\\, a_{\\sigma(2)},\\, \\ldots,\\, a_{\\sigma(m)}\\}\\)\uc5d0 \ud3ec\ud568\ub41c\ub2e4.
              \n\\[\\left|\\sum_{n=1}^m a_{\\sigma(n)} – \\sum_{n=1}^N a_n\\right| \\leq \\sum_{n=N+1}^{\\infty} |a_n| < \\varepsilon\\]\n\uc774\ubbc0\ub85c, \\(m \\to \\infty\\)\uc77c \ub54c \\(S' = S\\)\ub97c \uc5bb\ub294\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 7.11. (\ub9ac\ub9cc \uc7ac\ubc30\uc5f4 \uc815\ub9ac)<\/span><\/p>\n

              \ubb34\ud55c\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 \uc774 \ubb34\ud55c\uae09\uc218\uac00 \uc591\uc758 \ubb34\ud55c\ub300 \ub610\ub294 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub3c4\ub85d \uc7ac\ubc30\uc5f4\ud560 \uc218\ub3c4 \uc788\ub2e4.<\/p>\n<\/div>\n

              \n

              \uc99d\uba85<\/p>\n

              \uc591\uc218\uc778 \ud56d\uacfc \uc74c\uc218\uc778 \ud56d\uc744 \ubd84\ub9ac\ud558\uc5ec \uc0dd\uac01\ud558\uc790.<\/p>\n

              \\(p_n = \\max\\{a_n,\\, 0\\}\\), \\(q_n = \\max\\{-a_n,\\, 0\\}\\)\uc774\ub77c\uace0 \uc815\uc758\ud558\uc790. \uadf8\ub7ec\uba74 \\(a_n = p_n – q_n\\)\uc774\uace0 \\(|a_n| = p_n + q_n\\)\uc774\ub2e4.
              \n\\(\\sum a_n\\)\uc774 \uc218\ub834\ud558\uace0 \\(\\sum |a_n|\\)\uc774 \ubc1c\uc0b0\ud558\ubbc0\ub85c, \\(\\sum p_n\\)\uacfc \\(\\sum q_n\\) \ubaa8\ub450 \ubc1c\uc0b0\ud55c\ub2e4. \uc65c\ub0d0\ud558\uba74, \ub9cc\uc57d \ub458 \ub2e4 \uc218\ub834\ud558\uba74 \\(\\sum |a_n| = \\sum(p_n + q_n)\\)\ub3c4 \uc218\ub834\ud558\uac8c \ub418\uc5b4 \ubaa8\uc21c\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n

              \uc784\uc758\uc758 \uc2e4\uc218 \\(L\\)\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \ub2e4\uc74c\uacfc \uac19\uc774 \uc7ac\ubc30\uc5f4\uc744 \uad6c\uc131\ud55c\ub2e4.<\/p>\n