non-existence of universal series convergence criterion

There exist many criteria for examining the convergence and divergence of series with positive terms (see e.g. determining series convergence).  They all are sufficient but not necessary.  It has also been asked whether there would be any criterion which were both sufficient and necessary.  The famous mathematician Niels Henrik Abel took this question under consideration and proved the


There is no sequence

ϱ1,ϱ2,ϱ3, (1)

of positive numbers such that every series  a1+a2+a3+  of positive terms convergesPlanetmathPlanetmath when the condition

limnϱnan= 0

is true but diverges when it is false.

Proof.  Let’s assume that there is a sequence (1) having the both properties.  We infer that the series  1ϱ1+1ϱ2+1ϱ3+  is divergent because  limn(ϱn1ϱn)0.  The theorem on slower divergent series guarantees us another divergent series s1+s2+s3+ such that the ratio   sn:1ϱn=ϱnsn  tends to the limit 0 as  n.  But this limit result concerning the series  s1+s2+s3+  should mean, according to our assumptionPlanetmathPlanetmath, that the series is convergent.  The contradictionMathworldPlanetmathPlanetmath shows that the theorem holds.


  • 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III.2.  Mercatorin Kirjapaino Osakeyhtiö. Helsinki (1940).
Title non-existence of universal series convergence criterion
Canonical name NonexistenceOfUniversalSeriesConvergenceCriterion
Date of creation 2013-03-22 15:08:33
Last modified on 2013-03-22 15:08:33
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Theorem
Classification msc 40A05
Related topic SlowerDivergentSeries
Related topic SlowerConvergentSeries
Related topic ErnstLindelof