slower convergent series



a1+a2+a3+ (1)

is a converging series with positive , then one can always form another converging series


such that

limngnan= (2)

Proof.  Let S be the sum of (1),  Sn=a1+a2++an  the nth partial sum of (1) and  Rn+1=S-Sn=an+1+an+2+  the corresponding remainder term.  Then we have


We set

gn:=anRn+Rn+1=Rn-Rn+1n=1, 2, 3,

Then the series  g1+g2+g3+  fulfils the requirements in the theorem.  Its gn are positive.  Further, it converges because its nth partial sum is equal to R1-Rn+1 which tends to the limit  R1=S  as  n  since  Rn+10;  this implies also (2).

Title slower convergent series
Canonical name SlowerConvergentSeries
Date of creation 2013-03-22 15:08:24
Last modified on 2013-03-22 15:08:24
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Theorem
Classification msc 40A05
Related topic SlowerDivergentSeries
Related topic NonExistenceOfUniversalSeriesConvergenceCriterion