slower divergent series



a1+a2+a3+ (1)

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


with positive such that

limnsnan=0. (2)

Proof.   Let  Sn=a1+a2++an  be the nth partial sum of (1).  Then we have


We set  s1:=S1  and


for  n=2, 3, 4,  Then the of the series


apparently are positive.  This series is however divergent, because the sum of its n first is equal to Sn which grows without bound along with n since (1) diverges.  For this reason we also get the result (2).

Remark.Niels Henrik Abel has presented a simpler example on such series s1+s2+s3+:

Title slower divergent series
Canonical name SlowerDivergentSeries
Date of creation 2013-03-22 15:08:27
Last modified on 2013-03-22 15:08:27
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Theorem
Classification msc 40A05
Related topic SlowerConvergentSeries
Related topic NonExistenceOfUniversalSeriesConvergenceCriterion