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[parent] remainder term (Definition)

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$.



"remainder term" is owned by PrimeFan. [ owner history (2) ]
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See Also: sum of series

Other names:  remainder, tail of series
Keywords:  partial sum

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Cross-references: sum, convergent, complex, real, series, partial sum
There are 29 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 5343 times total.

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AMS MSC: 40-00 (Sequences, series, summability :: General reference works )
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