remainder term

Let $S_{n}$ be the $n^{\\mathrm{th}}$ partial sum of the series $a_{1}\\!+\\!a_{2}\\!+\\cdots$ with real or complex $a_{n}$ ( $n=1,\\,2,\\,\\ldots$ ).

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If the series is convergent with sum $S$ , then we call the $R_{n}:=S\\!-\\!S_{n}$ the $n^{\\mathrm{th}}$ remainder term or simply remainder of the series ( $n=1,\\,2,\\,\\ldots$ ). Then $\\lim_{n\\to\\infty}R_{n}=0$ .
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If there exists a number $s$ such that $\\lim_{n\\to\\infty}(s\\!-\\!S_{n})=0$ , then the series is convergent and its sum is $s$ .

单词	RemainderTerm
释义	remainder term Let $S_{n}$ be the $n^{\\mathrm{th}}$ partial sum of the series $a_{1}\\!+\\!a_{2}\\!+\\cdots$ with real or complex $a_{n}$ ( $n=1,\\,2,\\,\\ldots$ ). • If the series is convergent with sum $S$ , then we call the $R_{n}:=S\\!-\\!S_{n}$ the $n^{\\mathrm{th}}$ remainder term or simply remainder of the series ( $n=1,\\,2,\\,\\ldots$ ). Then $\\lim_{n\\to\\infty}R_{n}=0$ . • If there exists a number $s$ such that $\\lim_{n\\to\\infty}(s\\!-\\!S_{n})=0$ , then the series is convergent and its sum is $s$ .
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