remainder term
Let ${S}_{n}$ be the ${n}^{\mathrm{th}}$ partial sum of the series ${a}_{1}+{a}_{2}+\mathrm{\cdots}$ with real or complex ${a}_{n}$ ($n=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots}$).

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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 ($n=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots}$). Then ${lim}_{n\to \mathrm{\infty}}{R}_{n}=0$.

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If there exists a number $s$ such that ${lim}_{n\to \mathrm{\infty}}(s{S}_{n})=0$, then the series is convergent and its sum is $s$.
Title  remainder term 

Canonical name  RemainderTerm 
Date of creation  20130322 14:51:02 
Last modified on  20130322 14:51:02 
Owner  PrimeFan (13766) 
Last modified by  PrimeFan (13766) 
Numerical id  8 
Author  PrimeFan (13766) 
Entry type  Definition 
Classification  msc 4000 
Synonym  remainder 
Synonym  tail of series 
Related topic  SumOfSeries 