Extending tests for convergence of number series


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Extending tests for convergence of number series
Elijah Liflyand, Sergey Tikhonov, and Maria Zeltser September, 2012

Liflyand et al

Extending monotonicity

September, 2012 1 / 28

Goals and History
Main Goal:
Relax the monotonicity assumption for the sequence of terms of the series.

Liflyand et al

Extending monotonicity

September, 2012 2 / 28

Goals and History
Main Goal:
Relax the monotonicity assumption for the sequence of terms of the series.
In various well-known tests for convergence/divergence of number series



ak ,

(1)

k=1

with positive ak, monotonicity of the sequence of these ak is the basic assumption.

Liflyand et al

Extending monotonicity

September, 2012 2 / 28

Goals and History
Main Goal:
Relax the monotonicity assumption for the sequence of terms of the series.
In various well-known tests for convergence/divergence of number series



ak ,

(1)

k=1

with positive ak, monotonicity of the sequence of these ak is the basic assumption.
Such series are frequently called monotone series.

Liflyand et al

Extending monotonicity

September, 2012 2 / 28

Goals and History
Main Goal:
Relax the monotonicity assumption for the sequence of terms of the series.
In various well-known tests for convergence/divergence of number series



ak ,

(1)

k=1

with positive ak, monotonicity of the sequence of these ak is the basic assumption.
Such series are frequently called monotone series.
Tests by Abel, Cauchy, de la Vallee Poussin, Dedekind, Dirichlet, du Bois Reymond, Ermakov, Leibniz, Maclaurin, Olivier, Sapogov, Schl¨omilch are related to monotonicity.

Liflyand et al

Extending monotonicity

September, 2012 2 / 28

References
Books:

Liflyand et al

Extending monotonicity

September, 2012 3 / 28

References
Books: T.J.I’a Bromwich, Introduction to the Theory of Infinite Series, MacMillan, New York, 1965.

Liflyand et al

Extending monotonicity

September, 2012 3 / 28

References
Books: T.J.I’a Bromwich, Introduction to the Theory of Infinite Series, MacMillan, New York, 1965. K. Knopp, Theory and Application of Infinite Series, Blackie& Son Ltd., London-Glasgow, 1928.

Liflyand et al

Extending monotonicity

September, 2012 3 / 28

References
Books: T.J.I’a Bromwich, Introduction to the Theory of Infinite Series, MacMillan, New York, 1965. K. Knopp, Theory and Application of Infinite Series, Blackie& Son Ltd., London-Glasgow, 1928. G.M. Fichtengolz, Infinite Series: Rudiments, Gordon and Breach, New York, 1970.

Liflyand et al

Extending monotonicity

September, 2012 3 / 28

References
Books: T.J.I’a Bromwich, Introduction to the Theory of Infinite Series, MacMillan, New York, 1965. K. Knopp, Theory and Application of Infinite Series, Blackie& Son Ltd., London-Glasgow, 1928. G.M. Fichtengolz, Infinite Series: Rudiments, Gordon and Breach, New York, 1970. D.D. Bonar and M.J. Khoury, Real Infinite Series, MAA, Washington, DC, 2006.

Liflyand et al

Extending monotonicity

September, 2012 3 / 28

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Extending tests for convergence of number series