remainder term series

For any series

a1+a2+a3+ (1)

of real or complex terms aj one may interpret its m’th remainder term

Rm:=am+1+am+2+ (2)

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

Sm(n):=am+1+am+2++am+n  (n= 1,2,). (3)

If  m+n=k, then the kth partial sum of the original series (1) is

Sk=Sm+Sm(n). (4)

For a fixed m, the limit limkSk apparently exists iff the limit limnSm(n) exists.  Thus we can write the
Theorem.  The series (1) is convergentMathworldPlanetmathPlanetmath if and only if each remainder term series (2) is convergent.

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


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

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