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

If the series is convergent with sum $S$, then we call the difference $R_{n}:=S\!\!S_{n}$ 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 $s$.
Keywords:
partial sum
Related:
SumOfSeries
Synonym:
remainder, tail of series
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Reference
Type of Math Object:
Definition
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tail of a series by alozano ✓