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 $s$