{"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":"2021-10-13T18:09:39","modified_gmt":"2021-10-13T09:09:39","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\\[\\sum_{n=1}^{\\infty} c_n\\]
\n\uc758 \ud56d\uc758 \ubd80\ud638\uac00 \uad50\ub300\ub85c \ub098\ud0c0\ub098\uba74, \uc989 \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(c_n c_{n+1} < 0\\)\uc774\uba74, \uc704 \ubb34\ud55c\uae09\uc218\ub97c \uad50\ub300\uae09\uc218<\/span>(alternating series)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \uc591\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\left\\{ a_n \\right\\}\\)\uc774 \ub2e8\uc870\uac10\uc18c\ud558\uba70 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \ubb34\ud55c\uae09\uc218
\n\\[\\sum_{n=1}^{\\infty} (-1)^n a_n\\]
\n\uc758 \ubd80\ubd84\ud569\uc744 \\(S_n\\)\uc774\ub77c\uace0 \ud558\uc790. \uc989
\n\\[S_n = \\sum_{k=1}^n (-1)^k a_k\\]
\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
\n\\[\\begin{align}
\nS_{2(n+1)} &= S_{2n+2} \\\\[5pt]
\n&= S_{2n} + (-1)^{2n+1} a_{2n+1} + (-1)^{2n+2} a_{2n+2} \\\\[5pt]
\n&= S_{2n} – a_{2n+1} + a_{2n+2} \\\\[5pt]
\n&\\le S_{2n}
\n\\end{align}\\]
\n\uc774\uace0
\n\\[\\begin{align}
\nS_{2(n+1)+1}
\n&= S_{2n+1} + (-1)^{2n+2} a_{2n+2} + (-1)^{2n+3} a_{2n+3} \\\\[5pt]
\n&= S_{2n+1} + a_{2n+2} – a_{2n+3} \\\\[5pt]
\n&\\ge S_{2n+1}
\n\\end{align}\\]
\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\left\\{ S_{2n} \\right\\}\\)\uc740 \ub2e8\uc870\uac10\uc18c\ud558\uace0 \\(\\left\\{ S_{2n+1}\\right\\}\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud55c\ub2e4. \ub610\ud55c \uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[S_1 \\le S_{2n+1} \\le S_{2n} \\le S_2\\]
\n\uc774\ubbc0\ub85c, \\(\\left\\{ S_{2n} \\right\\}\\)\uacfc \\(\\left\\{ S_{2n+1}\\right\\}\\)\uc774 \ubaa8\ub450 \uc720\uacc4\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\left\\{ S_{2n} \\right\\}\\)\uacfc \\(\\left\\{ S_{2n+1}\\right\\}\\)\uc774 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4. \uadf8 \uadf9\ud55c\uc744 \uac01\uac01 \\(S,\\) \\(S ‘ \\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(n\\rightarrow\\infty\\)\uc77c \ub54c
\n\\[\\left\\lvert S_{2n+1} – S_{2n} \\right\\rvert = \\left\\lvert a_{2n+1} \\right\\rvert \\,\\rightarrow\\,0\\]
\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(S = S’ \\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\left\\{ S_n \\right\\}\\)\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n

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

\n

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

\uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \ud56d\uc774 \uc591\uc218\uc774\uace0 \ub2e8\uc870\uac10\uc18c\ud55c\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \ubb34\ud55c\uae09\uc218 \\[\\sum_{n=1}^{\\infty} (-1)^n a_n\\]\uc774 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\[\\lim_{n\\rightarrow\\infty} a_n = 0\\]\uc778 \uac83\uc774\ub2e4.\n<\/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 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud568\uc744 \uc27d\uac8c \ubcf4\uc77c \uc218 \uc788\ub2e4.
\n\\[\\sum_{n=1}^{\\infty} \\frac{(-1)^n}{n} ,\\quad \\sum_{n=1}^{\\infty} \\frac{(-1)^n}{\\sqrt{n}} .\\]
\n\ucc38\uace0\ub85c, \uc774 \ub450 \ubb34\ud55c\uae09\uc218\ub294 \uc808\ub300\uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc989 \uc774 \ub450 \ubb34\ud55c\uae09\uc218\ub294 \uc870\uac74\uc218\ub834\ud55c\ub2e4.\n<\/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 \\(x\\)\uc758 \uac12\uc758 \ubc94\uc704\ub97c \uad6c\ud574 \ubcf4\uc790.
\n\\[\\sum_{n=1}^{\\infty} \\frac{x^n}{n}.\\]
\n\ub9cc\uc57d \\(x=0\\)\uc774\uba74 \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\uac00 \uc790\uba85\ud558\uac8c \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x\\ne 0\\)\uc778 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud558\uc790.<\/p>\n

\\(a_n = \\frac{x^n}{n}\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
\n\\[\\lim_{n\\rightarrow\\infty} \\frac{\\lvert a_{n+1}\\rvert}{\\lvert a_n \\rvert} = \\lvert x \\rvert \\]
\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ube44 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\ub294 \\(\\lvert x \\rvert < 1\\)\uc77c \ub54c \uc808\ub300\uc218\ub834\ud558\uace0 \\(\\lvert x \\rvert > 1\\)\uc77c \ub54c \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

\\(x = 1\\)\uc77c \ub54c
\n\\[\\sum_{n=1}^{\\infty} \\frac{x^n}{n} = \\sum_{n=1}^{\\infty} \\frac{1}{n}\\]
\n\uc740 \\(p=1\\)\uc778 \\(p\\)-\uae09\uc218\uc774\ubbc0\ub85c \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n

\\(x = -1\\)\uc77c \ub54c
\n\\[\\sum_{n=1}^{\\infty} \\frac{x^n}{n} = \\sum_{n=1}^{\\infty} \\frac{(-1)^n}{n}\\]
\n\uc740 \uad50\ub300\uae09\uc218 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \uc218\ub834\ud55c\ub2e4.<\/p>\n

\uadf8\ub7ec\ubbc0\ub85c \uc8fc\uc5b4\uc9c4 \ubb34\ud55c\uae09\uc218\uac00 \uc218\ub834\ud558\ub3c4\ub85d \ud558\ub294 \\(x\\)\uc758 \ubc94\uc704\ub294 \\(-1 \\le x < 1\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n

\uc218\uc5f4 \\(\\left\\{ a_n \\right\\}\\)\uc774 \ub2e8\uc870\uac10\uc18c\ud558\uace0 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\uace0 \ud558\uc790. \ub2e4\uc74c\uacfc \uac19\uc740 \uad50\ub300\uae09\uc218\ub97c \uc0b4\ud3b4\ubcf4\uc790.
\n\\[S = \\sum_{k=1}^{\\infty} (-1)^k a_k .\\]
\n\uc774 \ubb34\ud55c\uae09\uc218\uc758 \ubd80\ubd84\ud569\uc744 \\(S_n\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\begin{align}
\n\\lvert S-S_n \\rvert
\n&= \\lvert a_{n+1} – a_{n+2} + a_{n+3} – a_{n+4} + – \\cdots \\rvert \\\\[5pt]
\n&= \\lvert a_{n+1} – (a_{n+2} – a_{n+3}) – (a_{n+4} – a_{n+4} ) – \\cdots \\rvert \\\\[5pt]
\n&\\le \\lvert a_{n+1} \\rvert .
\n\\end{align}\\]
\n\uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c\uacfc \uac19\uc740 \uad50\ub300\uae09\uc218\uc758 \uc624\ucc28\uc758 \ud55c\uacc4 \uacf5\uc2dd\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

\n

\ub530\ub984\uc815\ub9ac 2.4.2.<\/span><\/p>\n

\\[\\left\\lvert \\sum_{k=1}^{\\infty} (-1)^k a_k – S_n \\right\\rvert \\le \\left\\lvert a_{n+1} \\right\\rvert \\]\n<\/p><\/div>\n


\n<\/p>\n

<\/p>\n

<\/p>\n

\n