{"id":6689,"date":"2021-07-20T23:55:59","date_gmt":"2021-07-20T14:55:59","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6689"},"modified":"2026-09-27T17:31:28","modified_gmt":"2026-09-27T08:31:28","slug":"alternating-series","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/alternating-series\/","title":{"rendered":"\uad50\ub300\uae09\uc218"},"content":{"rendered":"
\n

\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 2\uc7a5 4\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.  (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n

\ubb34\ud55c\uae09\uc218
\n\\[
\n\\sum_{n=1}^{\\infty}c_n
\n\\]
\n\uc5d0\uc11c \uc774\uc6c3\ud55c \ub450 \ud56d\uc758 \ubd80\ud638\uac00 \uacc4\uc18d \ubc88\uac08\uc544 \ub098\ud0c0\ub098\uba74 \uc774 \uae09\uc218\ub97c \uad50\ub300\uae09\uc218<\/span>(alternating series)\ub77c\uace0 \ubd80\ub978\ub2e4. \ud2b9\ud788 \uc591\uc218\uc5f4 \\(\\left\\{a_n\\right\\}\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[
\n\\sum_{n=1}^{\\infty}(-1)^{n-1}a_n
\n=
\na_1-a_2+a_3-a_4+\\cdots
\n\\]
\n\ub294 \uad50\ub300\uae09\uc218\uc774\ub2e4. \ubaa8\ub4e0 \ud56d\uc758 \ubd80\ud638\ub97c \ubc18\ub300\ub85c \ud55c
\n\\[
\n\\sum_{n=1}^{\\infty}(-1)^na_n
\n\\]
\n\ub3c4 \ubb3c\ub860 \uad50\ub300\uae09\uc218\uc774\uba70, \ub450 \uae09\uc218\uc758 \uc218\ub834 \uc5ec\ubd80\ub294 \uac19\ub2e4.<\/p>\n

<\/p>\n

\uad50\ub300\uae09\uc218 \ud310\uc815\ubc95<\/h2>\n
\n

\uc815\ub9ac 2.4.1. (\uad50\ub300\uae09\uc218 \ud310\uc815\ubc95, Alternating Series Test)<\/span><\/p>\n

\\(\\left\\{a_n\\right\\}\\)\uc774 \uc591\uc218\uc5f4\uc774\uace0 \ub2e8\uc870\uac10\uc18c\ud55c\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \ubb34\ud55c\uae09\uc218
\n\\[
\n\\sum_{n=1}^{\\infty}(-1)^{n-1}a_n
\n\\]
\n\uc774 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740
\n\\[
\n\\lim_{n\\rightarrow\\infty}a_n=0
\n\\]
\n\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

\n

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

\uba3c\uc800 \uae09\uc218\uac00 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \uc77c\ubc18\ud56d \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec
\n\\[
\n(-1)^{n-1}a_n\\rightarrow0.
\n\\]
\n\ub530\ub77c\uc11c
\n\\[
\na_n
\n=
\n\\left|
\n(-1)^{n-1}a_n
\n\\right|
\n\\rightarrow0.
\n\\]
\n\uadf8\ub7ec\ubbc0\ub85c \\(a_n\\rightarrow0\\)\uc740 \ud544\uc694\uc870\uac74\uc774\ub2e4.<\/p>\n

\uc774\uc81c
\n\\[
\na_n\\rightarrow0
\n\\]
\n\uc774\ub77c\uace0 \uac00\uc815\ud558\uace0 \uae09\uc218\uac00 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uc790. \ubd80\ubd84\ud569\uc744
\n\\[
\nS_n
\n=
\n\\sum_{k=1}^{n}(-1)^{k-1}a_k
\n\\]
\n\ub77c\uace0 \ud558\uc790.<\/p>\n

\uc9dd\uc218 \ubc88\uc9f8 \ubd80\ubd84\ud569\uc5d0 \ub300\ud558\uc5ec
\n\\[
\n\\begin{aligned}
\nS_{2n+2}
\n&=
\nS_{2n}
\n+a_{2n+1}-a_{2n+2}\\\\
\n&\\ge
\nS_{2n}
\n\\end{aligned}
\n\\]
\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\left\\{S_{2n}\\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud55c\ub2e4.<\/p>\n

\ud55c\ud3b8 \ud640\uc218 \ubc88\uc9f8 \ubd80\ubd84\ud569\uc5d0 \ub300\ud558\uc5ec
\n\\[
\n\\begin{aligned}
\nS_{2n+3}
\n&=
\nS_{2n+1}
\n-a_{2n+2}+a_{2n+3}\\\\
\n&\\le
\nS_{2n+1}
\n\\end{aligned}
\n\\]
\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(\\left\\{S_{2n+1}\\right\\}\\)\uc740 \ub2e8\uc870\uac10\uc18c\ud55c\ub2e4.<\/p>\n

\ub610\ud55c
\n\\[
\nS_{2n}
\n\\le
\nS_{2n+1}
\n\\]
\n\uc774\uace0
\n\\[
\nS_2
\n\\le
\nS_{2n}
\n\\le
\nS_{2n+1}
\n\\le
\nS_1.
\n\\]
\n\ub530\ub77c\uc11c \\(\\left\\{S_{2n}\\right\\}\\)\uc740 \uc704\ub85c \uc720\uacc4\uc774\uace0, \\(\\left\\{S_{2n+1}\\right\\}\\)\uc740 \uc544\ub798\ub85c \uc720\uacc4\uc774\ub2e4. \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \ub450 \uc218\uc5f4\uc740 \uac01\uac01 \uc5b4\ub5a4 \uc2e4\uc218 \\(L\\), \\(M\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/p>\n

\uadf8\ub7f0\ub370
\n\\[
\nS_{2n+1}-S_{2n}=a_{2n+1}\\rightarrow0
\n\\]
\n\uc774\ubbc0\ub85c
\n\\[
\nM-L=0.
\n\\]
\n\ub530\ub77c\uc11c
\n\\[
\nL=M.
\n\\]
\n\uc9dd\uc218 \ubc88\uc9f8 \ubd80\ubd84\ud569\uacfc \ud640\uc218 \ubc88\uc9f8 \ubd80\ubd84\ud569\uc774 \uac19\uc740 \uac12\uc5d0 \uc218\ub834\ud558\ubbc0\ub85c \uc804\uccb4 \ubd80\ubd84\ud569\uc218\uc5f4 \\(\\left\\{S_n\\right\\}\\)\ub3c4 \uadf8 \uac12\uc5d0 \uc218\ub834\ud55c\ub2e4. \ub530\ub77c\uc11c \uc8fc\uc5b4\uc9c4 \uad50\ub300\uae09\uc218\ub294 \uc218\ub834\ud55c\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\n

\ubcf4\uae30 2.4.1.<\/span>
\n\uad50\ub300\uae09\uc218 \ud310\uc815\ubc95\uc744 \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c \ub450 \uae09\uc218\uac00 \uc218\ub834\ud568\uc744 \uc54c \uc218 \uc788\ub2e4.
\n\\[
\n\\sum_{n=1}^{\\infty}\\frac{(-1)^{n-1}}{n},
\n\\qquad
\n\\sum_{n=1}^{\\infty}\\frac{(-1)^{n-1}}{\\sqrt n}.
\n\\]\n<\/p>\n

\uc2e4\uc81c\ub85c
\n\\[
\n\\frac1n\\rightarrow0,
\n\\qquad
\n\\frac1{\\sqrt n}\\rightarrow0
\n\\]
\n\uc774\uace0 \ub450 \uc218\uc5f4\uc740 \ub2e8\uc870\uac10\uc18c\ud55c\ub2e4.<\/p>\n

\ud55c\ud3b8 \uc808\ub313\uac12\uc744 \ucde8\ud55c \uae09\uc218\ub294 \uac01\uac01
\n\\[
\n\\sum_{n=1}^{\\infty}\\frac1n,
\n\\qquad
\n\\sum_{n=1}^{\\infty}\\frac1{\\sqrt n}
\n\\]
\n\uc774\ub2e4. \uc774\ub4e4\uc740 \uac01\uac01 \\(p=1\\), \\(p=\\frac12\\)\uc778 \\(p\\)-\uae09\uc218\uc774\ubbc0\ub85c \ubaa8\ub450 \ubc1c\uc0b0\ud55c\ub2e4. \ub530\ub77c\uc11c \uc6d0\ub798 \ub450 \uae09\uc218\ub294 \ubaa8\ub450 \uc870\uac74\uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n

\n

\ubcf4\uae30 2.4.2.<\/span>
\n\ub2e4\uc74c \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud558\ub3c4\ub85d \ud558\ub294 \uc2e4\uc218 \\(x\\)\uc758 \ubc94\uc704\ub97c \uad6c\ud558\uc2dc\uc624.
\n\\[
\n\\sum_{n=1}^{\\infty}\\frac{x^n}{n}.
\n\\]\n<\/p>\n

\ud480\uc774.<\/span>
\n\uba3c\uc800
\n\\[
\n|x| < 1\n\\]\n\uc774\ub77c\uace0 \ud558\uc790. \uc808\ub313\uac12 \uae09\uc218\uc758 \uc77c\ubc18\ud56d\uc744\n\\[\nb_n=\\frac{|x|^n}{n}\n\\]\n\uc774\ub77c\uace0 \ud558\uba74\n\\[\n\\frac{b_{n+1}}{b_n}\n=\n|x|\\frac{n}{n+1}\n\\rightarrow\n|x| < 1.\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\uc774 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc6d0\ub798 \uae09\uc218\ub294 \uc808\ub300\uc218\ub834\ud55c\ub2e4.<\/p>\n

\uc774\uc81c
\n\\[
\n|x|>1
\n\\]
\n\uc774\ub77c\uace0 \ud558\uc790. \uc774 \uacbd\uc6b0
\n\\[
\n\\frac{|x|^{n+1}\/(n+1)}{|x|^n\/n}
\n=
\n|x|\\frac{n}{n+1}
\n\\rightarrow
\n|x|>1.
\n\\]
\n\ub530\ub77c\uc11c \\(|x|^n\/n\\)\uc740 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
\n\\[
\n\\frac{x^n}{n}
\n\\]
\n\ub3c4 \\(0\\)\uc5d0 \uc218\ub834\ud558\uc9c0 \uc54a\uc73c\uba70, \uc77c\ubc18\ud56d \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \uc8fc\uc5b4\uc9c4 \uae09\uc218\ub294 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

\\(x=1\\)\uc774\uba74
\n\\[
\n\\sum_{n=1}^{\\infty}\\frac{x^n}{n}
\n=
\n\\sum_{n=1}^{\\infty}\\frac1n
\n\\]
\n\uc774\ubbc0\ub85c \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

\\(x=-1\\)\uc774\uba74
\n\\[
\n\\sum_{n=1}^{\\infty}\\frac{x^n}{n}
\n=
\n\\sum_{n=1}^{\\infty}\\frac{(-1)^n}{n}.
\n\\]
\n\uc774 \uae09\uc218\ub294 \ubaa8\ub4e0 \ud56d\uc758 \ubd80\ud638\ub97c \ubc14\uafb8\uba74
\n\\[
\n\\sum_{n=1}^{\\infty}\\frac{(-1)^{n-1}}{n}
\n\\]
\n\uc774 \ub418\ubbc0\ub85c \uad50\ub300\uae09\uc218 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \uc218\ub834\ud55c\ub2e4.<\/p>\n

\ub530\ub77c\uc11c \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740
\n\\[
\n-1\\le x < 1\n\\]\n\uc774\ub2e4.<\/p>\n<\/div>\n

<\/p>\n

\uad50\ub300\uae09\uc218\uc758 \uc624\ucc28 \ucd94\uc815<\/h2>\n

\\(\\left\\{a_n\\right\\}\\)\uc774 \uc591\uc218\uc5f4\uc774\uace0 \ub2e8\uc870\uac10\uc18c\ud558\uba70
\n\\[
\na_n\\rightarrow0
\n\\]
\n\uc774\ub77c\uace0 \ud558\uc790. \uad50\ub300\uae09\uc218
\n\\[
\nS
\n=
\n\\sum_{k=1}^{\\infty}(-1)^{k-1}a_k
\n\\]
\n\uc758 \\(n\\)\ubc88\uc9f8 \ubd80\ubd84\ud569\uc744
\n\\[
\nS_n
\n=
\n\\sum_{k=1}^{n}(-1)^{k-1}a_k
\n\\]
\n\ub77c\uace0 \ud558\uc790.<\/p>\n

\uc815\ub9ac 2.4.1\uc758 \uc99d\uba85\uc5d0\uc11c
\n\\[
\nS_{2m}\\le S\\le S_{2m+1}
\n\\]
\n\uc784\uc744 \uc54c \uc218 \uc788\ub2e4.<\/p>\n

\\(n\\)\uc774 \uc9dd\uc218\uc774\uba74
\n\\[
\nS_n\\le S\\le S_{n+1}
\n=
\nS_n+a_{n+1},
\n\\]
\n\ub530\ub77c\uc11c
\n\\[
\n0\\le S-S_n\\le a_{n+1}.
\n\\]\n<\/p>\n

\\(n\\)\uc774 \ud640\uc218\uc774\uba74
\n\\[
\nS_{n+1}\\le S\\le S_n
\n\\]
\n\uc774\uace0
\n\\[
\nS_n-S_{n+1}=a_{n+1}
\n\\]
\n\uc774\ubbc0\ub85c
\n\\[
\n0\\le S_n-S\\le a_{n+1}.
\n\\]\n<\/p>\n

\ub450 \uacbd\uc6b0\ub97c \ud569\uce58\uba74 \ub2e4\uc74c \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

\n

\ub530\ub984\uc815\ub9ac 2.4.2. (\uad50\ub300\uae09\uc218\uc758 \uc624\ucc28 \ucd94\uc815)<\/span><\/p>\n

\\(\\left\\{a_n\\right\\}\\)\uc774 \uc591\uc218\uc5f4\uc774\uace0 \ub2e8\uc870\uac10\uc18c\ud558\uba70
\n\\[
\na_n\\rightarrow0
\n\\]
\n\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c
\n\\[
\nS
\n=
\n\\sum_{k=1}^{\\infty}(-1)^{k-1}a_k,
\n\\qquad
\nS_n
\n=
\n\\sum_{k=1}^{n}(-1)^{k-1}a_k
\n\\]
\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
\n\\[
\n|S-S_n|
\n\\le
\na_{n+1}.
\n\\]
\n\uc989 \uad50\ub300\uae09\uc218\ub97c \\(n\\)\ubc88\uc9f8 \ud56d\uae4c\uc9c0 \uacc4\uc0b0\ud588\uc744 \ub54c \uc0dd\uae30\ub294 \uc624\ucc28\uc758 \uc808\ub313\uac12\uc740 \uccab \ubc88\uc9f8 \uc0dd\ub7b5\ub41c \ud56d\uc758 \uc808\ub313\uac12\ubcf4\ub2e4 \ud06c\uc9c0 \uc54a\ub2e4.<\/p>\n<\/div>\n


\n<\/p>\n

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<\/p>\n

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