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 $m$’th remainder term

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

as a series.  This remainder term series has its own partial 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 ${lim}_{k\to \mathrm{\infty }}{S}_k$ apparently exists 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 if each remainder term series (2) is convergent.

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

Title	remainder term series
Canonical name	RemainderTermSeries
Date of creation	2014-05-16 21:09:46
Last modified on	2014-05-16 21:09:46
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	13
Author	pahio (2872)
Entry type	Definition
Classification	msc 40-00