{"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":"2026-09-27T17:25:21","modified_gmt":"2026-09-27T08:25:21","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\\[
\n\\sum_{n=1}^{\\infty}a_n
\n\\]
\n\uc744 \uc774 \ucc45\uc5d0\uc11c\ub294 \uc591\ud56d\uae09\uc218<\/span>(series of nonnegative terms)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc591\ud56d\uae09\uc218\uc5d0\uc11c\ub294 \ubd80\ubd84\ud569\uc774 \ub2e8\uc870\uc99d\uac00\ud558\ubbc0\ub85c \uc218\uc5f4\uc758 \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uc5ec \uc5ec\ub7ec \uac00\uc9c0 \uc218\ub834 \ud310\uc815\ubc95\uc744 \uc5bb\uc744 \uc218 \uc788\ub2e4.<\/p>\n

\uae09\uc218\uc758 \uc2dc\uc791 \ucca8\uc790\ub294 \ubc18\ub4dc\uc2dc \\(1\\)\uc77c \ud544\uc694\ub294 \uc5c6\ub2e4. \uc608\ub97c \ub4e4\uc5b4
\n\\[
\n\\sum_{n=0}^{\\infty}a_n
\n\\]
\n\uacfc \uac19\uc740 \uae09\uc218\ub3c4 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \ub610\ud55c \uc720\ud55c \uac1c\uc758 \ud56d\uc744 \ucd94\uac00\ud558\uac70\ub098 \uc81c\uac70\ud574\ub3c4 \uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub294 \ubcc0\ud558\uc9c0 \uc54a\ub294\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\uace0
\n\\[
\nS_n=\\sum_{k=1}^{n}a_k
\n\\]
\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
\n\\[
\nS_{n+1}=S_n+a_{n+1}\\ge S_n
\n\\]
\n\uc774\ubbc0\ub85c \ubd80\ubd84\ud569\uc218\uc5f4 \\(\\left\\{S_n\\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud55c\ub2e4.<\/p>\n

\ub530\ub77c\uc11c \\(\\left\\{S_n\\right\\}\\)\uc774 \uc704\ub85c \uc720\uacc4\uc774\uba74 \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uc218\ub834\ud55c\ub2e4. \ubc18\ub300\ub85c \\(\\left\\{S_n\\right\\}\\)\uc774 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\uba74 \ub2e8\uc870\uc99d\uac00\uc218\uc5f4\uc774\ubbc0\ub85c
\n\\[
\nS_n\\rightarrow\\infty.
\n\\]
\n\uc774\ub85c\ubd80\ud130 \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\\[
\n\\sum_{n=1}^{\\infty}a_n
\n\\]
\n\uc774 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ubd80\ubd84\ud569\uc218\uc5f4 \\(\\left\\{S_n\\right\\}\\)\uc774 \uc704\ub85c \uc720\uacc4\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

\n

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

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

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

    (1) \\(n\\ge2\\)\uc77c \ub54c
    \n\\[
    \n\\frac{1}{n^2}
    \n\\le
    \n\\frac{1}{(n-1)n}
    \n=
    \n\\frac{1}{n-1}-\\frac{1}{n}.
    \n\\]
    \n\ub530\ub77c\uc11c
    \n\\[
    \n\\begin{aligned}
    \nS_n
    \n&=
    \n1+\\sum_{k=2}^{n}\\frac{1}{k^2}\\\\
    \n&\\le
    \n1+
    \n\\sum_{k=2}^{n}
    \n\\left(
    \n\\frac{1}{k-1}-\\frac{1}{k}
    \n\\right)\\\\
    \n&=
    \n2-\\frac1n\\\\
    \n&\\le2.
    \n\\end{aligned}
    \n\\]
    \n\ub530\ub77c\uc11c \ubd80\ubd84\ud569\uc218\uc5f4\uc740 \uc704\ub85c \uc720\uacc4\uc774\uace0, \uc720\uacc4 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec
    \n\\[
    \n\\sum_{n=1}^{\\infty}\\frac1{n^2}
    \n\\]
    \n\uc740 \uc218\ub834\ud55c\ub2e4.<\/p>\n

    (2) \\(0!=1\\)\ub85c \uc815\uc758\ud55c\ub2e4. \\(k\\ge2\\)\uc774\uba74
    \n\\[
    \nk!\\ge2^{k-1}
    \n\\]
    \n\uc774\ubbc0\ub85c
    \n\\[
    \n\\frac1{k!}\\le\\frac1{2^{k-1}}.
    \n\\]
    \n\ub530\ub77c\uc11c
    \n\\[
    \n\\begin{aligned}
    \nS_n
    \n&=
    \n\\sum_{k=0}^{n}\\frac1{k!}\\\\
    \n&\\le
    \n1+1+
    \n\\frac12+\\frac1{2^2}+\\cdots+\\frac1{2^{n-1}}\\\\
    \n&<\n3.\n\\end{aligned}\n\\]\n\uadf8\ub7ec\ubbc0\ub85c \ubd80\ubd84\ud569\uc218\uc5f4\uc774 \uc704\ub85c \uc720\uacc4\uc774\uace0\n\\[\n\\sum_{n=0}^{\\infty}\\frac1{n!}\n\\]\n\uc740 \uc218\ub834\ud55c\ub2e4.<\/p>\n

    (3) \\(m\\)\uc774 \uc591\uc758 \uc815\uc218\ub77c\uace0 \ud558\uc790. \ubd80\ubd84\ud569 \\(S_{2^m}\\)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ubb36\uc744 \uc218 \uc788\ub2e4.
    \n\\[
    \n\\begin{aligned}
    \nS_{2^m}
    \n&=
    \n1+\\frac12
    \n+
    \n\\left(\\frac13+\\frac14\\right)
    \n+
    \n\\left(\\frac15+\\frac16+\\frac17+\\frac18\\right)
    \n+\\cdots\\\\
    \n&\\qquad+
    \n\\left(
    \n\\frac1{2^{m-1}+1}
    \n+\\cdots+
    \n\\frac1{2^m}
    \n\\right)\\\\
    \n&\\ge
    \n1+\\frac12
    \n+
    \n2\\cdot\\frac14
    \n+
    \n4\\cdot\\frac18
    \n+\\cdots+
    \n2^{m-1}\\cdot\\frac1{2^m}\\\\
    \n&=
    \n1+\\frac{m}{2}.
    \n\\end{aligned}
    \n\\]
    \n\\(m\\)\uc744 \uc5bc\ub9c8\ub4e0\uc9c0 \ud06c\uac8c \ud560 \uc218 \uc788\uc73c\ubbc0\ub85c \ubd80\ubd84\ud569\uc218\uc5f4\uc740 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ub2e4. \ub530\ub77c\uc11c
    \n\\[
    \n\\sum_{n=1}^{\\infty}\\frac1n
    \n\\]
    \n\uc740 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n<\/div>\n

    <\/p>\n

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

    \uc608\uc81c 2.2.1\uc758 (2)\uc5d0\uc11c \ubb34\ud55c\uae09\uc218
    \n\\[
    \n\\sum_{n=0}^{\\infty}\\frac1{n!}
    \n\\]
    \n\uc774 \uc218\ub834\ud568\uc744 \ubcf4\uc600\ub2e4. \uc774 \uae09\uc218\uc758 \ud569\uc744 \uc790\uc5f0\uc0c1\uc218<\/span>(Euler’s number)\ub77c\uace0 \ubd80\ub974\uace0
    \n\\[
    \ne
    \n\\]
    \n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989
    \n\\[
    \ne
    \n=
    \n\\sum_{n=0}^{\\infty}\\frac1{n!}
    \n=
    \n2.718281828459045\\cdots.
    \n\\]\n<\/p>\n

    \uc790\uc5f0\uc0c1\uc218\ub294 \uc218\uc5f4\uc758 \uadf9\ud55c\uc73c\ub85c\ub3c4 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \ub2e4\uc74c \uc218\uc5f4\uc744 \uc0dd\uac01\ud558\uc790.
    \n\\[
    \ne_n
    \n=
    \n\\left(
    \n1+\\frac1n
    \n\\right)^n.
    \n\\]
    \n\uc774\ud56d \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uba74
    \n\\[
    \ne_n
    \n=
    \n2+
    \n\\sum_{k=2}^{n}
    \n\\frac1{k!}
    \n\\left(
    \n1-\\frac1n
    \n\\right)
    \n\\left(
    \n1-\\frac2n
    \n\\right)
    \n\\cdots
    \n\\left(
    \n1-\\frac{k-1}{n}
    \n\\right).
    \n\\]\n<\/p>\n

    \\(2\\le k\\le n\\)\uc77c \ub54c \uac01 \uc778\uc218\ub294 \\(n\\)\uc744 \\(n+1\\)\ub85c \ubc14\uafb8\uba74 \ucee4\uc9c4\ub2e4. \ub610\ud55c \\(e_{n+1}\\)\uc5d0\ub294 \\(e_n\\)\uc5d0 \uc5c6\ub294 \uc591\uc758 \ud56d\uc774 \ud558\ub098 \ub354 \uc788\uc73c\ubbc0\ub85c
    \n\\[
    \ne_n < e_{n+1}.\n\\]\n\ub530\ub77c\uc11c \\(\\left\\{e_n\\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud55c\ub2e4.<\/p>\n

    \ud55c\ud3b8 \uac01 \uad04\ud638 \uc548\uc758 \uc778\uc218\ub294 \\(1\\) \uc774\ud558\uc774\ubbc0\ub85c
    \n\\[
    \n\\begin{aligned}
    \ne_n
    \n&\\le
    \n\\sum_{k=0}^{n}\\frac1{k!}\\\\
    \n&\\le
    \n\\sum_{k=0}^{\\infty}\\frac1{k!}
    \n=
    \ne.
    \n\\end{aligned}
    \n\\]
    \n\ub530\ub77c\uc11c \\(\\left\\{e_n\\right\\}\\)\uc740 \uc704\ub85c \uc720\uacc4\uc774\uace0 \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uc5b4\ub5a4 \uc2e4\uc218 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/p>\n

    \uc774\uc81c \\(L=e\\)\uc784\uc744 \ubcf4\uc774\uc790. \uc591\uc758 \uc815\uc218 \\(m\\)\uc744 \uace0\uc815\ud558\uace0 \\(n\\ge m\\)\uc774\ub77c\uace0 \ud558\uba74
    \n\\[
    \ne_n
    \n\\ge
    \n2+
    \n\\sum_{k=2}^{m}
    \n\\frac1{k!}
    \n\\left(
    \n1-\\frac1n
    \n\\right)
    \n\\left(
    \n1-\\frac2n
    \n\\right)
    \n\\cdots
    \n\\left(
    \n1-\\frac{k-1}{n}
    \n\\right).
    \n\\]
    \n\\(n\\rightarrow\\infty\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74
    \n\\[
    \nL
    \n\\ge
    \n\\sum_{k=0}^{m}\\frac1{k!}.
    \n\\]
    \n\uc774 \ubd80\ub4f1\uc2dd\uc740 \ubaa8\ub4e0 \\(m\\)\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud558\ubbc0\ub85c \\(m\\rightarrow\\infty\\)\ub85c \ubcf4\ub0b4\uba74
    \n\\[
    \nL\\ge e.
    \n\\]
    \n\ud55c\ud3b8 \\(e_n\\le e\\)\uc774\ubbc0\ub85c \\(L\\le e\\)\uc774\ub2e4. \ub530\ub77c\uc11c
    \n\\[
    \nL=e.
    \n\\]\n<\/p>\n

    \n

    \uc790\uc5f0\uc0c1\uc218<\/span>
    \n\\[
    \n{
    \n\\lim_{n\\rightarrow\\infty}
    \n\\left(
    \n1+\\frac1n
    \n\\right)^n
    \n=
    \n\\sum_{n=0}^{\\infty}\\frac1{n!}
    \n=
    \ne
    \n}
    \n\\]\n<\/p>\n<\/div>\n

    \\(e>1\\)\uc774\ubbc0\ub85c \ubc11\uc774 \\(e\\)\uc778 \ub85c\uadf8\ud568\uc218\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uc774\ub97c \uc790\uc5f0\ub85c\uadf8<\/span>(natural logarithm)\ub77c\uace0 \ubd80\ub974\uace0
    \n\\[
    \n\\ln x
    \n=
    \n\\log_e x
    \n\\qquad
    \n(x>0)
    \n\\]
    \n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

    <\/p>\n

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

    \ubcf4\uc870\uc815\ub9ac. (\uc720\ud55c \uac1c\uc758 \ud56d\uacfc \uae09\uc218\uc758 \uc218\ub834)<\/span><\/p>\n

    \ubb34\ud55c\uae09\uc218\uc5d0\uc11c \uc720\ud55c \uac1c\uc758 \ud56d\uc744 \ucd94\uac00\ud558\uac70\ub098 \uc81c\uac70\ud558\uac70\ub098 \ubc14\uafb8\uc5b4\ub3c4 \uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub294 \ubcc0\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85.<\/p>\n

    \uc720\ud55c \uac1c\uc758 \ud56d\uc744 \ucd94\uac00\ud558\uac70\ub098 \uc81c\uac70\ud558\uba74 \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0\uc11c \ub450 \ubd80\ubd84\ud569\uc218\uc5f4\uc758 \ucc28\uc774\ub294 \ud558\ub098\uc758 \uc0c1\uc218\uac00 \ub41c\ub2e4. \uc0c1\uc218\ub97c \ub354\ud558\uac70\ub098 \ube7c\ub294 \uac83\uc740 \uc218\uc5f4\uc758 \uc218\ub834 \uc5ec\ubd80\uc5d0 \uc601\ud5a5\uc744 \uc8fc\uc9c0 \uc54a\uc73c\ubbc0\ub85c \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \uc774\uc81c \\(\\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, \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \na_n\\le b_n
    \n\\]
    \n\uc774\ub77c\uace0 \ud558\uc790. \\(N\\) \uc774\ud6c4\uc5d0\uc11c \uc774 \ubd80\ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4\uba74
    \n\\[
    \n\\sum_{k=N}^{n}a_k
    \n\\le
    \n\\sum_{k=N}^{n}b_k.
    \n\\]
    \n\uc774\ub97c \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c \ud310\uc815\ubc95\uc744 \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, \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \na_n\\le b_n
    \n\\]
    \n\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

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

      \uc99d\uba85.<\/p>\n

      (1) \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0\uc11c \\(a_n\\le b_n\\)\uc774\ub77c\uace0 \ud558\uc790. \\(\\sum b_n\\)\uc774 \uc218\ub834\ud558\uba74 \uadf8 \uaf2c\ub9ac \ubd80\ubd84\ud569\uc740 \uc704\ub85c \uc720\uacc4\uc774\ub2e4. \ub530\ub77c\uc11c
      \n\\[
      \n\\sum_{k=N}^{n}a_k
      \n\\le
      \n\\sum_{k=N}^{n}b_k
      \n\\]
      \n\uc5d0 \uc758\ud558\uc5ec \\(\\sum a_n\\)\uc758 \uaf2c\ub9ac \ubd80\ubd84\ud569\ub3c4 \uc704\ub85c \uc720\uacc4\uc774\ub2e4. \uc720\uacc4 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \uaf2c\ub9ac \uae09\uc218\uac00 \uc218\ub834\ud558\uace0, \uc720\ud55c \uac1c\uc758 \ucc98\uc74c \ud56d\uc740 \uc218\ub834 \uc5ec\ubd80\uc5d0 \uc601\ud5a5\uc744 \uc8fc\uc9c0 \uc54a\uc73c\ubbc0\ub85c \\(\\sum a_n\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n

      (2)\ub294 (1)\uc758 \ub300\uc6b0\uc774\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      \n

      \uc608\uc81c 2.2.2.<\/span>
      \n\\(p\\ge2\\)\ub77c\uace0 \ud558\uc790. \uc608\uc81c 2.2.1\uc758 (1)\uc5d0 \uc758\ud558\uc5ec
      \n\\[
      \n\\sum_{n=1}^{\\infty}\\frac1{n^2}
      \n\\]
      \n\uc740 \uc218\ub834\ud55c\ub2e4. \ub610\ud55c
      \n\\[
      \n0\\le
      \n\\frac1{n^p}
      \n\\le
      \n\\frac1{n^2}.
      \n\\]
      \n\ub530\ub77c\uc11c \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec
      \n\\[
      \n\\sum_{n=1}^{\\infty}\\frac1{n^p}
      \n\\]
      \n\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n

      \n

      \uc608\uc81c 2.2.3.<\/span>
      \n\uc608\uc81c 2.2.1\uc758 (3)\uc5d0 \uc758\ud558\uc5ec
      \n\\[
      \n\\sum_{n=1}^{\\infty}\\frac1n
      \n\\]
      \n\uc740 \ubc1c\uc0b0\ud55c\ub2e4. \uadf8\ub7f0\ub370
      \n\\[
      \n\\frac1n
      \n\\le
      \n\\frac1{\\sqrt n}
      \n\\]
      \n\uc774\ubbc0\ub85c \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec
      \n\\[
      \n\\sum_{n=1}^{\\infty}\\frac1{\\sqrt n}
      \n\\]
      \n\ub3c4 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n<\/div>\n

      <\/p>\n

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

      \\(\\left\\{a_n\\right\\}\\)\uacfc \\(\\left\\{b_n\\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \uc591\uc218\uc774\uace0
      \n\\[
      \n\\lim_{n\\rightarrow\\infty}\\frac{a_n}{b_n}
      \n=
      \n\\rho,
      \n\\qquad
      \n0 < \\rho < \\infty\n\\]\n\ub77c\uace0 \ud558\uc790.<\/p>\n

      \uc608\ub97c \ub4e4\uc5b4
      \n\\[
      \n\\frac{\\rho}{2} < \\rho < \\frac{3\\rho}{2}\n\\]\n\uc774\ubbc0\ub85c \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\n\\frac{\\rho}{2}\n<\n\\frac{a_n}{b_n}\n<\n\\frac{3\\rho}{2}.\n\\]\n\ub530\ub77c\uc11c\n\\[\n\\frac{\\rho}{2}b_n\n<\na_n\n<\n\\frac{3\\rho}{2}b_n.\n\\]\n\ube44\uad50 \ud310\uc815\ubc95\uc744 \uc591\ucabd\uc5d0 \uc801\uc6a9\ud558\uba74 \ub450 \uae09\uc218\ub294 \ub3d9\uc2dc\uc5d0 \uc218\ub834\ud558\uac70\ub098 \ub3d9\uc2dc\uc5d0 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

      \n

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

      \\(\\left\\{a_n\\right\\}\\)\uacfc \\(\\left\\{b_n\\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \uc591\uc218\uc774\uace0
      \n\\[
      \n\\lim_{n\\rightarrow\\infty}
      \n\\frac{a_n}{b_n}
      \n=
      \n\\rho,
      \n\\qquad
      \n0 < \\rho < \\infty\n\\]\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74\n\\[\n\\sum_{n=1}^{\\infty}a_n\n\\]\n\uc774 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740\n\\[\n\\sum_{n=1}^{\\infty}b_n\n\\]\n\uc774 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4.<\/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\ud558\uc2dc\uc624.
      \n\\[
      \n\\sum_{n=1}^{\\infty}
      \n\\frac{4n^2-n+3}{n^3+2n}.
      \n\\]\n<\/p>\n

      \ud480\uc774.<\/span>
      \n\ubd84\ubaa8\uc758 \ucc28\uc218\uac00 \ubd84\uc790\uc758 \ucc28\uc218\ubcf4\ub2e4 \\(1\\) \ud06c\ub2e4\ub294 \uc0ac\uc2e4\uc5d0 \uc8fc\ubaa9\ud558\uc5ec
      \n\\[
      \na_n
      \n=
      \n\\frac{4n^2-n+3}{n^3+2n},
      \n\\qquad
      \nb_n=\\frac1n
      \n\\]
      \n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
      \n\\[
      \n\\begin{aligned}
      \n\\frac{a_n}{b_n}
      \n&=
      \n\\frac{4n^3-n^2+3n}{n^3+2n}\\\\
      \n&\\rightarrow4.
      \n\\end{aligned}
      \n\\]
      \n\uadf8\ub7f0\ub370
      \n\\[
      \n\\sum_{n=1}^{\\infty}\\frac1n
      \n\\]
      \n\uc740 \ubc1c\uc0b0\ud558\ubbc0\ub85c \uadf9\ud55c \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec
      \n\\[
      \n\\sum_{n=1}^{\\infty}
      \n\\frac{4n^2-n+3}{n^3+2n}
      \n\\]
      \n\ub3c4 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n<\/div>\n

      <\/p>\n

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

      \ube44\uad50 \ud310\uc815\ubc95\uacfc \uadf9\ud55c \ube44\uad50 \ud310\uc815\ubc95\uc744 \uc0ac\uc6a9\ud558\ub824\uba74 \uc218\ub834 \uc5ec\ubd80\uac00 \uc774\ubbf8 \uc54c\ub824\uc9c4 \ub2e4\ub978 \uae09\uc218\uc640 \ube44\uad50\ud574\uc57c \ud55c\ub2e4. \ub2e4\uc74c\uc758 \ube44 \ud310\uc815\ubc95\uacfc \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc740 \uc8fc\uc5b4\uc9c4 \uc77c\ubc18\ud56d \uc790\uccb4\ub97c \uc774\uc6a9\ud558\uc5ec \uc218\ub834 \uc5ec\ubd80\ub97c \uc870\uc0ac\ud55c\ub2e4.<\/p>\n

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

      \uba3c\uc800
      \n\\[
      \n\\rho < 1\n\\]\n\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc2e4\uc218 \\(r\\)\ub97c \ud0dd\ud560 \uc218 \uc788\ub2e4.\n\\[\n\\rho < r < 1.\n\\]\n\uadf8\ub7ec\uba74 \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\n\\frac{a_{n+1}}{a_n}\n\\le r,\n\\]\n\uc989\n\\[\na_{n+1}\\le ra_n.\n\\]\n\uc5b4\ub5a4 \\(N\\) \uc774\ud6c4\uc5d0 \uc774 \ubd80\ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4\uba74\n\\[\na_{N+k}\n\\le\nr^k a_N\n\\qquad\n(k\\ge1).\n\\]\n\ub530\ub77c\uc11c \uae09\uc218\uc758 \uaf2c\ub9ac\ub294 \uc218\ub834\ud558\ub294 \ub4f1\ube44\uae09\uc218\uc5d0 \uc758\ud558\uc5ec \uc704\uc5d0\uc11c \uc5b5\uc81c\ub418\ubbc0\ub85c \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec\n\\[\n\\sum_{n=1}^{\\infty}a_n\n\\]\n\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n

      \uc774\uc81c
      \n\\[
      \n\\rho>1
      \n\\]
      \n\uc774\ub77c\uace0 \ud558\uc790. \\(1 < r < \\rho\\)\uc778 \uc2e4\uc218 \\(r\\)\ub97c \ud0dd\ud558\uba74 \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\n\\frac{a_{n+1}}{a_n}>r.
      \n\\]
      \n\ub530\ub77c\uc11c
      \n\\[
      \na_{N+k}
      \n\\ge
      \nr^k a_N,
      \n\\]
      \n\uc774\ubbc0\ub85c \\(a_n\\)\uc740 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc77c\ubc18\ud56d \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \uae09\uc218\ub294 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

      \n

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

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

        \n
      1. \\(\\rho < 1\\)\uc774\uba74\n\\[\n\\sum_{n=1}^{\\infty}a_n\n\\]\n\uc740 \uc218\ub834\ud55c\ub2e4.<\/li>\n
      2. \\(\\rho>1\\)\uc774\uba74
        \n\\[
        \n\\sum_{n=1}^{\\infty}a_n
        \n\\]
        \n\uc740 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n
      3. \\(\\rho=1\\)\uc774\uba74 \uc774 \ud310\uc815\ubc95\ub9cc\uc73c\ub85c\ub294 \uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \uacb0\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\ud558\uc2dc\uc624.
        \n\\[
        \n\\sum_{n=1}^{\\infty}\\frac{x^n}{n!}.
        \n\\]\n<\/p>\n

        \ud480\uc774.<\/span>
        \n\\[
        \na_n=\\frac{x^n}{n!}
        \n\\]
        \n\uc774\ub77c\uace0 \ud558\uba74
        \n\\[
        \n\\frac{a_{n+1}}{a_n}
        \n=
        \n\\frac{x}{n+1}
        \n\\rightarrow0.
        \n\\]
        \n\ub530\ub77c\uc11c \ube44 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec
        \n\\[
        \n\\sum_{n=1}^{\\infty}\\frac{x^n}{n!}
        \n\\]
        \n\uc740 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc608\uc81c 2.2.6.<\/span>
        \n\ub450 \ubb34\ud55c\uae09\uc218
        \n\\[
        \n\\sum_{n=1}^{\\infty}\\frac1n,
        \n\\qquad
        \n\\sum_{n=1}^{\\infty}\\frac1{n^2}
        \n\\]
        \n\uc744 \uc0dd\uac01\ud558\uc790. \uccab\uc9f8 \uae09\uc218\ub294 \ubc1c\uc0b0\ud558\uace0 \ub458\uc9f8 \uae09\uc218\ub294 \uc218\ub834\ud55c\ub2e4.<\/p>\n

        \uadf8\ub7f0\ub370
        \n\\[
        \n\\lim_{n\\rightarrow\\infty}
        \n\\frac{\\frac1{n+1}}{\\frac1n}
        \n=
        \n1
        \n\\]
        \n\uc774\uace0
        \n\\[
        \n\\lim_{n\\rightarrow\\infty}
        \n\\frac{\\frac1{(n+1)^2}}{\\frac1{n^2}}
        \n=
        \n1.
        \n\\]
        \n\ub530\ub77c\uc11c \ube44 \ud310\uc815\ubc95\uc5d0\uc11c \\(\\rho=1\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \uae09\uc218\uac00 \uc218\ub834\ud560 \uc218\ub3c4 \uc788\uace0 \ubc1c\uc0b0\ud560 \uc218\ub3c4 \uc788\ub2e4.<\/p>\n<\/div>\n

        <\/p>\n

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

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

        \uba3c\uc800
        \n\\[
        \n\\rho < 1\n\\]\n\uc774\ub77c\uace0 \ud558\uc790. \\(\\rho < r < 1\\)\uc778 \uc2e4\uc218 \\(r\\)\ub97c \ud0dd\ud558\uba74 \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\n\\sqrt[n]{a_n}\\le r.\n\\]\n\ub530\ub77c\uc11c\n\\[\na_n\\le r^n.\n\\]\n\ub4f1\ube44\uae09\uc218\n\\[\n\\sum r^n\n\\]\n\uc774 \uc218\ub834\ud558\ubbc0\ub85c \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec\n\\[\n\\sum a_n\n\\]\n\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/p>\n

        \ub2e4\uc74c\uc73c\ub85c
        \n\\[
        \n\\rho>1
        \n\\]
        \n\uc774\ub77c\uace0 \ud558\uc790. \\(1 < r < \\rho\\)\uc778 \\(r\\)\ub97c \ud0dd\ud558\uba74 \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(n\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\n\\sqrt[n]{a_n}>r.
        \n\\]
        \n\ub530\ub77c\uc11c
        \n\\[
        \na_n>r^n,
        \n\\]
        \n\uc774\ubbc0\ub85c \\(a_n\\)\uc740 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \ub530\ub77c\uc11c \uae09\uc218\ub294 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

        \n

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

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

          \n
        1. \\(\\rho < 1\\)\uc774\uba74\n\\[\n\\sum_{n=1}^{\\infty}a_n\n\\]\n\uc740 \uc218\ub834\ud55c\ub2e4.<\/li>\n
        2. \\(\\rho>1\\)\uc774\uba74
          \n\\[
          \n\\sum_{n=1}^{\\infty}a_n
          \n\\]
          \n\uc740 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n
        3. \\(\\rho=1\\)\uc774\uba74 \uc774 \ud310\uc815\ubc95\ub9cc\uc73c\ub85c\ub294 \uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \uacb0\uc815\ud560 \uc218 \uc5c6\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
          \n

          \uc608\uc81c 2.2.7.<\/span>
          \n\\(x\\ge0\\)\uc77c \ub54c \ub2e4\uc74c \ubb34\ud55c\uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \ud310\uc815\ud558\uc2dc\uc624.
          \n\\[
          \n1+2x+x^2+2x^3+x^4+2x^5+\\cdots.
          \n\\]\n<\/p>\n

          \ud480\uc774.<\/span>
          \n\\(x=0\\)\uc774\uba74 \uae09\uc218\ub294 \uc790\uba85\ud558\uac8c \uc218\ub834\ud55c\ub2e4. \uc774\uc81c \\(x>0\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

          \uccab\uc9f8 \ud56d\uc744 \\(a_1\\)\ub85c \ub193\uc73c\uba74
          \n\\[
          \na_n
          \n=
          \n\\begin{cases}
          \nx^{n-1}, & n\\text{\uc774 \ud640\uc218\uc77c \ub54c},\\\\[4pt]
          \n2x^{n-1}, & n\\text{\uc774 \uc9dd\uc218\uc77c \ub54c}.
          \n\\end{cases}
          \n\\]
          \n\ub530\ub77c\uc11c
          \n\\[
          \n\\frac{a_{n+1}}{a_n}
          \n\\]
          \n\uc740 \\(2x\\)\uc640 \\(x\/2\\)\uac00 \ubc88\uac08\uc544 \ub098\ud0c0\ub098\ubbc0\ub85c \uc77c\ubc18\uc801\uc73c\ub85c \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4. \ub530\ub77c\uc11c \ube44 \ud310\uc815\ubc95\uc744 \uc9c1\uc811 \uc801\uc6a9\ud560 \uc218 \uc5c6\ub2e4.<\/p>\n

          \ud55c\ud3b8 \\(n\\)\uc774 \ud640\uc218\uc774\uba74
          \n\\[
          \n\\sqrt[n]{a_n}
          \n=
          \nx^{(n-1)\/n}
          \n\\rightarrow x
          \n\\]
          \n\uc774\uace0, \\(n\\)\uc774 \uc9dd\uc218\uc774\uba74
          \n\\[
          \n\\sqrt[n]{a_n}
          \n=
          \n2^{1\/n}x^{(n-1)\/n}
          \n\\rightarrow x.
          \n\\]
          \n\ub530\ub77c\uc11c
          \n\\[
          \n\\lim_{n\\rightarrow\\infty}
          \n\\sqrt[n]{a_n}
          \n=
          \nx.
          \n\\]
          \n\uadf8\ub7ec\ubbc0\ub85c \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(x < 1\\)\uc774\uba74 \uae09\uc218\uac00 \uc218\ub834\ud558\uace0 \\(x>1\\)\uc774\uba74 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

          \\(x=1\\)\uc774\uba74 \uc77c\ubc18\ud56d\uc774
          \n\\[
          \n1,\\,2,\\,1,\\,2,\\ldots
          \n\\]
          \n\uc640 \uac19\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c \uae09\uc218\ub294 \ubc1c\uc0b0\ud55c\ub2e4. \ub530\ub77c\uc11c \uc8fc\uc5b4\uc9c4 \uae09\uc218\ub294
          \n\\[
          \n0\\le x < 1\n\\]\n\uc77c \ub54c\uc5d0\ub9cc \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n

          \ucc38\uace0.<\/span>
          \n\uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc5d0\uc11c \uadf9\ud55c\uc774 \\(1\\)\uc774\uba74 \ud310\uc815\ud560 \uc218 \uc5c6\ub294 \uacbd\uc6b0\uac00 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4
          \n\\[
          \n\\sqrt[n]{\\frac1n}\\rightarrow1,
          \n\\qquad
          \n\\sqrt[n]{\\frac1{n^2}}\\rightarrow1
          \n\\]
          \n\uc774\uc9c0\ub9cc
          \n\\[
          \n\\sum\\frac1n
          \n\\]
          \n\uc740 \ubc1c\uc0b0\ud558\uace0
          \n\\[
          \n\\sum\\frac1{n^2}
          \n\\]
          \n\uc740 \uc218\ub834\ud55c\ub2e4.<\/p>\n

          <\/p>\n

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

          \\(\\left\\{a_n\\right\\}\\)\uc774 \uc74c\uc774 \uc544\ub2cc \ud56d\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ub2e8\uc870\uac10\uc18c\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c \ub450 \uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \ube44\uad50\ud560 \uc218 \uc788\ub2e4.
          \n\\[
          \n\\sum_{n=1}^{\\infty}a_n,
          \n\\qquad
          \n\\sum_{k=1}^{\\infty}2^k a_{2^k}.
          \n\\]\n<\/p>\n

          \n

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

          \\(\\left\\{a_n\\right\\}\\)\uc774 \uc74c\uc774 \uc544\ub2cc \ud56d\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ub2e8\uc870\uac10\uc18c\uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
          \n\\[
          \n\\sum_{n=1}^{\\infty}a_n
          \n\\]
          \n\uc774 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740
          \n\\[
          \n\\sum_{k=1}^{\\infty}2^k a_{2^k}
          \n\\]
          \n\uc774 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

          \n

          \uc99d\uba85.<\/p>\n

          \uba3c\uc800
          \n\\[
          \n\\sum_{n=1}^{\\infty}a_n
          \n\\]
          \n\uc774 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \ub2e8\uc870\uac10\uc18c\uc131\uc5d0 \uc758\ud558\uc5ec
          \n\\[
          \na_{2^{k-1}+1},\\ldots,a_{2^k}
          \n\\]
          \n\uc758 \uac01 \ud56d\uc740 \\(a_{2^k}\\) \uc774\uc0c1\uc774\ub2e4. \ub530\ub77c\uc11c
          \n\\[
          \n2^{k-1}a_{2^k}
          \n\\le
          \n\\sum_{j=2^{k-1}+1}^{2^k}a_j.
          \n\\]
          \n\uadf8\ub7ec\ubbc0\ub85c
          \n\\[
          \n\\frac12
          \n\\sum_{k=1}^{m}2^ka_{2^k}
          \n\\le
          \n\\sum_{j=2}^{2^m}a_j.
          \n\\]
          \n\uc624\ub978\ucabd\uc774 \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c \uc67c\ucabd\ub3c4 \uc704\ub85c \uc720\uacc4\uc774\ub2e4. \uc720\uacc4 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec
          \n\\[
          \n\\sum_{k=1}^{\\infty}2^ka_{2^k}
          \n\\]
          \n\uc740 \uc218\ub834\ud55c\ub2e4.<\/p>\n

          \uac70\uafb8\ub85c
          \n\\[
          \n\\sum_{k=1}^{\\infty}2^ka_{2^k}
          \n\\]
          \n\uc774 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \ub2e8\uc870\uac10\uc18c\uc131\uc5d0 \uc758\ud558\uc5ec
          \n\\[
          \n\\sum_{j=2^k}^{2^{k+1}-1}a_j
          \n\\le
          \n2^k a_{2^k}.
          \n\\]
          \n\ub530\ub77c\uc11c
          \n\\[
          \n\\begin{aligned}
          \nS_{2^{m+1}-1}
          \n&=
          \na_1+
          \n\\sum_{k=1}^{m}
          \n\\sum_{j=2^k}^{2^{k+1}-1}a_j\\\\
          \n&\\le
          \na_1+
          \n\\sum_{k=1}^{m}2^ka_{2^k}\\\\
          \n&\\le
          \na_1+
          \n\\sum_{k=1}^{\\infty}2^ka_{2^k}.
          \n\\end{aligned}
          \n\\]
          \n\ubd80\ubd84\ud569\uc218\uc5f4\uc740 \ub2e8\uc870\uc99d\uac00\ud558\uace0, \uc704\uc758 \ubd80\ubd84\ud569\ub4e4\uc5d0 \uc758\ud558\uc5ec \uc804\uccb4\uac00 \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c \uc720\uacc4 \ud310\uc815\ubc95\uc5d0 \ub530\ub77c
          \n\\[
          \n\\sum_{n=1}^{\\infty}a_n
          \n\\]
          \n\uc774 \uc218\ub834\ud55c\ub2e4.
          \n<\/span><\/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

            \n
          1. \\(\\displaystyle\\sum_{n=1}^{\\infty}\\frac1n\\)<\/li>\n
          2. \\(\\displaystyle\\sum_{n=2}^{\\infty}\\frac1{n\\ln n}\\)<\/li>\n
          3. \\(\\displaystyle\\sum_{n=2}^{\\infty}\\frac1{n(\\ln n)^2}\\)<\/li>\n
          4. \\(\\displaystyle\\sum_{n=9}^{\\infty}\\frac1{n\\ln(\\ln n)}\\)<\/li>\n<\/ol>\n

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

            (1) \uc218\uc5f4 \\(\\left\\{1\/n\\right\\}\\)\uc740 \ub2e8\uc870\uac10\uc18c\ud55c\ub2e4. \uc751\uc9d1\ud55c \uae09\uc218\ub294
            \n\\[
            \n\\sum_{k=1}^{\\infty}
            \n2^k\\frac1{2^k}
            \n=
            \n\\sum_{k=1}^{\\infty}1
            \n\\]
            \n\uc774\ubbc0\ub85c \ubc1c\uc0b0\ud55c\ub2e4. \ub530\ub77c\uc11c
            \n\\[
            \n\\sum_{n=1}^{\\infty}\\frac1n
            \n\\]
            \n\ub3c4 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

            (2) \\(n\\ge2\\)\uc5d0\uc11c
            \n\\[
            \n\\frac1{n\\ln n}
            \n\\]
            \n\uc740 \uc591\uc218\uc774\uace0 \ub2e8\uc870\uac10\uc18c\ud55c\ub2e4. \uc720\ud55c \uac1c\uc758 \ud56d\uc740 \uc218\ub834 \uc5ec\ubd80\uc5d0 \uc601\ud5a5\uc744 \uc8fc\uc9c0 \uc54a\uc73c\ubbc0\ub85c \uc751\uc9d1 \ud310\uc815\ubc95\uc744 \uc801\uc6a9\ud558\uba74
            \n\\[
            \n\\begin{aligned}
            \n\\sum
            \n2^k\\frac1{2^k\\ln(2^k)}
            \n&=
            \n\\sum\\frac1{k\\ln2}\\\\
            \n&=
            \n\\frac1{\\ln2}
            \n\\sum\\frac1k.
            \n\\end{aligned}
            \n\\]
            \n\uc774\ub294 \ubc1c\uc0b0\ud558\ubbc0\ub85c
            \n\\[
            \n\\sum_{n=2}^{\\infty}\\frac1{n\\ln n}
            \n\\]
            \n\ub3c4 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

            (3) \ub9c8\ucc2c\uac00\uc9c0\ub85c
            \n\\[
            \n\\begin{aligned}
            \n\\sum
            \n2^k\\frac1{2^k(\\ln(2^k))^2}
            \n&=
            \n\\sum
            \n\\frac1{k^2(\\ln2)^2}\\\\
            \n&=
            \n\\frac1{(\\ln2)^2}
            \n\\sum\\frac1{k^2}.
            \n\\end{aligned}
            \n\\]
            \n\ub9c8\uc9c0\ub9c9 \uae09\uc218\ub294 \uc218\ub834\ud558\ubbc0\ub85c
            \n\\[
            \n\\sum_{n=2}^{\\infty}
            \n\\frac1{n(\\ln n)^2}
            \n\\]
            \n\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/p>\n

            (4) \\(n\\ge9\\)\uc5d0\uc11c
            \n\\[
            \n\\frac1{n\\ln(\\ln n)}
            \n\\]
            \n\uc740 \uc591\uc218\uc774\uace0 \ub2e8\uc870\uac10\uc18c\ud55c\ub2e4. \ucda9\ubd84\ud788 \ud070 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \uc751\uc9d1\ud55c \uc77c\ubc18\ud56d\uc740
            \n\\[
            \n2^k
            \n\\frac1{
            \n2^k\\ln(\\ln2^k)
            \n}
            \n=
            \n\\frac1{\\ln(k\\ln2)}.
            \n\\]
            \n\uadf8\ub7f0\ub370 \ucda9\ubd84\ud788 \ud070 \\(k\\)\uc5d0 \ub300\ud558\uc5ec
            \n\\[
            \n0 < \\ln(k\\ln2) < k\n\\]\n\uc774\ubbc0\ub85c\n\\[\n\\frac1{\\ln(k\\ln2)}\n>
            \n\\frac1k.
            \n\\]
            \n\uc870\ud654\uae09\uc218\uac00 \ubc1c\uc0b0\ud558\ubbc0\ub85c \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \uc751\uc9d1\ud55c \uae09\uc218\ub3c4 \ubc1c\uc0b0\ud55c\ub2e4. \ub530\ub77c\uc11c
            \n\\[
            \n\\sum_{n=9}^{\\infty}
            \n\\frac1{n\\ln(\\ln n)}
            \n\\]
            \n\uc740 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n<\/div>\n

            \n

            \uc608\uc81c 2.2.9.<\/span>
            \n\\(p\\)\uac00 \uc2e4\uc218\uc77c \ub54c \ubb34\ud55c\uae09\uc218
            \n\\[
            \n\\sum_{n=1}^{\\infty}\\frac1{n^p}
            \n\\]
            \n\ub97c \\(\\boldsymbol{p}\\)-\uae09\uc218<\/span>(\\(p\\)-series)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \uae09\uc218\uac00 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc774
            \n\\[
            \np>1
            \n\\]
            \n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n

            \ud480\uc774.<\/span>
            \n\\(p\\le0\\)\uc774\uba74
            \n\\[
            \n\\frac1{n^p}
            \n\\]
            \n\uac00 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c \uc77c\ubc18\ud56d \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \uae09\uc218\ub294 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

            \uc774\uc81c \\(p>0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc218\uc5f4
            \n\\[
            \n\\left\\{\\frac1{n^p}\\right\\}
            \n\\]
            \n\uc740 \ub2e8\uc870\uac10\uc18c\ud55c\ub2e4. \ucf54\uc2dc\uc758 \uc751\uc9d1 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \uc6d0\ub798 \uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub294
            \n\\[
            \n\\begin{aligned}
            \n\\sum_{n=1}^{\\infty}
            \n2^n\\frac1{(2^n)^p}
            \n&=
            \n\\sum_{n=1}^{\\infty}
            \n2^{n(1-p)}\\\\
            \n&=
            \n\\sum_{n=1}^{\\infty}
            \n\\left(
            \n2^{1-p}
            \n\\right)^n
            \n\\end{aligned}
            \n\\]
            \n\uc758 \uc218\ub834 \uc5ec\ubd80\uc640 \uac19\ub2e4.<\/p>\n

            \ub9c8\uc9c0\ub9c9 \uae09\uc218\ub294 \ubb34\ud55c\ub4f1\ube44\uae09\uc218\uc774\uba70 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740
            \n\\[
            \n2^{1-p} < 1\n\\]\n\uc774\ub2e4. \ubc11 \\(2\\)\uc778 \uc9c0\uc218\ud568\uc218\ub294 \uc99d\uac00\ud558\ubbc0\ub85c \uc774\ub294\n\\[\n1-p < 0,\n\\]\n\uc989\n\\[\np>1
            \n\\]
            \n\uacfc \ub3d9\uce58\uc774\ub2e4. \ub530\ub77c\uc11c
            \n\\[
            \n{
            \n\\sum_{n=1}^{\\infty}\\frac1{n^p}
            \n\\text{\uc774 \uc218\ub834\ud55c\ub2e4}
            \n\\quad\\Longleftrightarrow\\quad
            \np>1
            \n}
            \n\\]
            \n\uc774\ub2e4.<\/p>\n<\/div>\n

            <\/p>\n

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

            \uc55e\uc758 \uc81c\uacf1\uadfc \ud310\uc815\ubc95\uc740
            \n\\[
            \n\\sqrt[n]{a_n}
            \n\\]
            \n\uc758 \uadf9\ud55c\uc774 \uc874\uc7ac\ud560 \ub54c\uc5d0\ub9cc \uc801\uc6a9\ud560 \uc218 \uc788\ub2e4. \uc0c1\uadf9\ud55c\uc744 \uc774\uc6a9\ud558\uba74 \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294 \uacbd\uc6b0\uc5d0\ub3c4 \uac19\uc740 \uc544\uc774\ub514\uc5b4\ub97c \uc801\uc6a9\ud560 \uc218 \uc788\ub2e4.<\/p>\n

            \\(\\left\\{a_n\\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \\(0\\) \uc774\uc0c1\uc774\uace0
            \n\\[
            \n\\rho
            \n=
            \n\\varlimsup_{n\\rightarrow\\infty}
            \n\\sqrt[n]{a_n}
            \n\\]
            \n\ub77c\uace0 \ud558\uc790. \uc5ec\uae30\uc11c \\(\\rho\\)\ub294 \\(\\infty\\)\uc77c \uc218\ub3c4 \uc788\ub2e4.<\/p>\n

            \uba3c\uc800
            \n\\[
            \n\\rho < 1\n\\]\n\uc774\ub77c\uace0 \ud558\uc790. \\(\\rho < r < 1\\)\uc778 \uc2e4\uc218 \\(r\\)\ub97c \ud0dd\ud55c\ub2e4. \uc55e \uc808\uc758 \uc0c1\uadf9\ud55c \uc815\uc758\uc5d0 \uc758\ud558\uc5ec\n\\[\n\\sup_{k\\ge n}\\sqrt[k]{a_k}\n\\rightarrow\\rho.\n\\]\n\ub530\ub77c\uc11c \ucda9\ubd84\ud788 \ud070 \\(n\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\n\\sup_{k\\ge n}\\sqrt[k]{a_k}\n<\nr.\n\\]\n\ud2b9\ud788 \ucda9\ubd84\ud788 \ud070 \ubaa8\ub4e0 \\(k\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\n\\sqrt[k]{a_k}\n<\nr,\n\\]\n\uc989\n\\[\na_k\n

            \uc774\ubc88\uc5d0\ub294
            \n\\[
            \n\\rho>1
            \n\\]
            \n\uc774\ub77c\uace0 \ud558\uc790. \\(1 < r < \\rho\\)\uc778 \uc2e4\uc218 \\(r\\)\ub97c \ud0dd\ud55c\ub2e4. \\(\\rho=\\infty\\)\uc778 \uacbd\uc6b0\uc5d0\ub3c4 \uc774\ub7ec\ud55c \\(r\\)\ub97c \ud0dd\ud560 \uc218 \uc788\ub2e4.<\/p>\n

            \uc0c1\uadf9\ud55c\uc774 \\(r\\)\ubcf4\ub2e4 \ud06c\ubbc0\ub85c \uc784\uc758\uc758 \\(N\\)\uc5d0 \ub300\ud558\uc5ec \uc5b4\ub5a4 \\(n\\ge N\\)\uc774 \uc874\uc7ac\ud558\uc5ec
            \n\\[
            \n\\sqrt[n]{a_n}>r
            \n\\]
            \n\uc774\ub2e4. \ub530\ub77c\uc11c \uc774\ub7ec\ud55c \\(n\\)\uc774 \ubb34\ud55c\ud788 \ub9ce\uc774 \uc874\uc7ac\ud558\uace0
            \n\\[
            \na_n>r^n>1
            \n\\]
            \n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(a_n\\)\uc740 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc77c\ubc18\ud56d \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec
            \n\\[
            \n\\sum a_n
            \n\\]
            \n\uc740 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

            \n

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

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

              \n
            1. \\(\\rho < 1\\)\uc774\uba74\n\\[\n\\sum_{n=1}^{\\infty}a_n\n\\]\n\uc740 \uc218\ub834\ud55c\ub2e4.<\/li>\n
            2. \\(\\rho>1\\)\uc774\uba74
              \n\\[
              \n\\sum_{n=1}^{\\infty}a_n
              \n\\]
              \n\uc740 \ubc1c\uc0b0\ud55c\ub2e4.<\/li>\n
            3. \\(\\rho=1\\)\uc774\uba74 \uc774 \ud310\uc815\ubc95\ub9cc\uc73c\ub85c\ub294 \uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub97c \uacb0\uc815\ud560 \uc218 \uc5c6\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

              \ubcf4\ud1b5\uc740 \uc774 \uc0c1\uadf9\ud55c \ud615\ud0dc\uae4c\uc9c0 \ud3ec\ud568\ud558\uc5ec \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
              \n\\[
              \na_{2n}
              \n=
              \n\\frac1{n^n},
              \n\\qquad
              \na_{2n-1}
              \n=
              \n\\frac1{2^n}
              \n\\]
              \n\uc73c\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc989
              \n\\[
              \n\\frac12,\\,
              \n1,\\,
              \n\\frac1{2^2},\\,
              \n\\frac1{2^2},\\,
              \n\\frac1{2^3},\\,
              \n\\frac1{3^3},\\,
              \n\\frac1{2^4},\\,
              \n\\frac1{4^4},\\,
              \n\\ldots
              \n\\]
              \n\uc640 \uac19\uc740 \uc218\uc5f4\uc774\ub2e4.<\/p>\n

              \uc9dd\uc218 \ubc88\uc9f8 \ud56d\uc5d0 \ub300\ud574\uc11c\ub294
              \n\\[
              \n\\sqrt[2n]{a_{2n}}
              \n=
              \n\\frac1{\\sqrt n}
              \n\\rightarrow0
              \n\\]
              \n\uc774\uace0, \ud640\uc218 \ubc88\uc9f8 \ud56d\uc5d0 \ub300\ud574\uc11c\ub294
              \n\\[
              \n\\sqrt[2n-1]{a_{2n-1}}
              \n=
              \n\\frac1{2^{\\,n\/(2n-1)}}
              \n\\rightarrow
              \n\\frac1{\\sqrt2}.
              \n\\]
              \n\ub530\ub77c\uc11c
              \n\\[
              \n\\varlimsup_{n\\rightarrow\\infty}
              \n\\sqrt[n]{a_n}
              \n=
              \n\\frac1{\\sqrt2}
              \n< 1.\n\\]\n\uadf8\ub7ec\ubbc0\ub85c \uc81c\uacf1\uadfc \uc0c1\uadf9\ud55c \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec\n\\[\n\\sum_{n=1}^{\\infty}a_n\n\\]\n\uc740 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n

              <\/p>\n

              <\/p>\n

              \n