remainder term series

For any series

$a_1+a_2+a_3+\mathrm{\dots }$(1)

of real or complex terms $a_j$ one may interpret its $m$’thremainder term

$R_m:=a_{m+1}+a_{m+2}+\mathrm{\dots }$(2)

as a series.  This remainder term series has its ownpartial sums

$S_m^{(n)}:=a_{m+1}+a_{m+2}+\mathrm{\dots }+a_{m+n}\mathit{\hspace{1em}\hspace{1em}}(n=\mathrm{\hspace{0.33em}1},2,\mathrm{\dots }).$(3)

If  $m+n=k$, then the $k^{\mathrm{th}}$ partial sum of the original series (1)is

$S_k=S_m+S_m^{(n)}.$(4)

For a fixed $m$, the limit ${lim}_{k\to \mathrm{\infty }}{S}_k$ apparentlyexists iff the limit${lim}_{n\to \mathrm{\infty }}{S}_m^{(n)}$ exists.  Thus we can write the
Theorem.  The series (1) is convergent if and only ifeach remainder term series (2) is convergent.

Cf. the entry ‘‘finite changes in convergent series’’.