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 $s$ .