# remainder term series

For any series

 $\displaystyle a_{1}+a_{2}+a_{3}+\ldots$ (1)

of real or complex terms $a_{j}$ one may interpret its $m$’th remainder term

 $\displaystyle R_{m}\;:=\;a_{m+1}+a_{m+2}+\ldots$ (2)

as a series.  This remainder term series has its own partial sums

 $\displaystyle S_{m}^{(n)}\;:=\;a_{m+1}+a_{m+2}+\ldots+a_{m+n}\qquad(n\;=\;1,2,% \ldots).$ (3)

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

 $\displaystyle S_{k}\;=\;S_{m}+S_{m}^{(n)}.$ (4)

For a fixed $m$, the limit $\lim_{k\to\infty}S_{k}$ apparently exists iff the limit $\lim_{n\to\infty}S_{m}^{(n)}$ 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’’.

## References

• 1 Л. Д. Кудрявцев: Математический анализ. Издательство  ‘‘Высшая  школа’’. Москва (1970).

Title remainder term series RemainderTermSeries 2014-05-16 21:09:46 2014-05-16 21:09:46 pahio (2872) pahio (2872) 13 pahio (2872) Definition msc 40-00