# remainder term

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

• 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  $\lim_{n\to\infty}R_{n}=0$.

• 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 $s$.

Title remainder term RemainderTerm 2013-03-22 14:51:02 2013-03-22 14:51:02 PrimeFan (13766) PrimeFan (13766) 8 PrimeFan (13766) Definition msc 40-00 remainder tail of series SumOfSeries