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remainder term
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(Definition)
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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 $s$
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"remainder term" is owned by PrimeFan. [ owner history (2) ]
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Cross-references: number, sum, convergent, complex, real, series, partial sum
There are 23 references to this entry.
This is version 5 of remainder term, born on 2004-11-24, modified 2006-09-29.
Object id is 6522, canonical name is RemainderTerm.
Accessed 7528 times total.
Classification:
| AMS MSC: | 40-00 (Sequences, series, summability :: General reference works ) |
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Pending Errata and Addenda
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