if @\slimits@@@k=1ak converges then ak0

Theorem 1.

Suppose a1,a2, is a sequencePlanetmathPlanetmath of real or complex numbersMathworldPlanetmathPlanetmath. If the series


converges, then limkak=0.


  1. 1.

    The harmonic series k=11/k shows that the implicationMathworldPlanetmath can not be reversed.

  2. 2.

    This result can be used as a first test for convergence of a series k=1ak. If ak does not converge to 0, then k=1ak does not converge either.


Let S be the value of the sum, and let ε>0 be arbitrary. Then there exists an N1 such that


for all MN. For jN we then have

|aj+1| = |k=1j+1ak-k=1jak|
< ε,

and the claim follows. ∎

Title if @\slimits@@@k=1ak converges then ak0
Canonical name Ifsumk1inftyAkConvergesThenAkto0
Date of creation 2013-03-22 15:00:38
Last modified on 2013-03-22 15:00:38
Owner matte (1858)
Last modified by matte (1858)
Numerical id 13
Author matte (1858)
Entry type Theorem
Classification msc 40-00
Synonym necessary condition of convergence
Related topic DeterminingSeriesConvergence
Related topic CompleteUltrametricField
Related topic ConvergenceConditionOfInfiniteProduct
Related topic LambertSeries
Related topic AbsoluteConvergenceOfIntegralAndBoundednessOfDerivative
Related topic ConvergentSeriesWhereNotOnlyA_nButAlsoNa_nTendsTo0