finite changes in convergent series

The following theorem means that at the beginning of a convergent seriesMathworldPlanetmathPlanetmath, one can remove or attach a finite amount of terms without influencing on the convergence of the series – the convergence is determined alone by the infinitely long “tail” of the series.  Consequently, one can also freely change the of a finite amount of terms.

Theorem.  Let k be a natural numberMathworldPlanetmath.  A series n=1an convergesPlanetmathPlanetmath iff the series n=k+1an converges.  Then the sums of both series are with

n=k+1an=n=1an-n=1kan. (1)

Proof.  Denote the kth partial sum of n=1an by Sk and the nth partial sum of n=k+1an by Sn.  Then we have

Sn=n=k+1k+nan=Sk+n-Sk. (2)

1.  If n=1an converges, i.e.  limnSn:=S  exists as a finite number, then (2) implies


Thus n=k+1an converges and (1) is true.

2.  If we suppose n=k+1an to be convergentMathworldPlanetmath, i.e.  limnSn:=S  exists as finite, then (2) implies that


This means that n=1an is convergent and  S=Sk+S,  which is (1), is in .

Title finite changes in convergent series
Canonical name FiniteChangesInConvergentSeries
Date of creation 2013-03-22 19:03:10
Last modified on 2013-03-22 19:03:10
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Theorem
Classification msc 40A05
Related topic SumOfSeriesDependsOnOrder
Related topic RiemannSeriesTheorem
Related topic RatioTestOfDAlembert