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 $a_n$ ($n=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots }$).

• 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  ${lim}_{n\to \mathrm{\infty }}{R}_n=0$.

• 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 $s$.