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Convergence of means

The following result due to Cauchy  [1]  , p. 59, started a series of important results in the convergence theory:

Theorem: If

(1)

is a sequence that converges to , then also the sequence of arithmetic means

(2)

converges to 0.

The result can be easily extended [2] to the form:

Theorem: If , then also

More generally Stolz [3] proved

Theorem: If and is a sequence of positive real numbers such that  

then

Finally Toeplitz [4]  proved the following generalization:

Theorem: Let , and let the numbers from the triangular scheme

(3)

satisfy the conditions

• the numbers of each column form a zero sequencers, that is for every we have with ,
• there exists a such that for every we have .

Then

Theorem:  Let the numbers from the triangular scheme (3) satisfy the conditions

• the numbers of each column form a zero sequencers, that is for every we have with ,
• there exists a such that for every we have ,
• for .

If , then also

Theorem:  Let the numbers from the triangular scheme (3) satisfy the conditions

• the numbers of each column form a zero sequencers, that is for every we have with ,
• there exists a such that for every we have ,
• for
• the elements on the diagonal form a zero sequences, that is for every we have for  .

If and  ,then

For geometric means we have [5] :

Theorem: If and for all , then for the sequence of geometric means we have

References

[1]	Cauchy, A. L. (1821). Cours d'Analyse de l'École Royale Polytechnique: Première Partie: Analyse Algébrique . Paris: Chez Debure frères.

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