Abstract
The well-known Leibniz Criterion or alternating series test of convergence of alternating series is generalized for the case when the absolute value of terms of series are “not absolutely monotonously” convergent to zero. Questions of accuracy of the estimation for the series remainder are considered.
G. Zverkina—Author expresses gratitude to Professor V. N. Chubarikov (Department of mechanics and mathematics of Lomonosov Moscow State University) and Department of mathematics of the Yaroslavl State Technical University, the organizer of the International student’s competition on mathematics in 2012.
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Notes
- 1.
Z-very-monotonously.
- 2.
Non-strict monotony means: \(\varphi (x)\) it is monotonous on \(\mathfrak {D}\), if \( \forall a <b \in \mathfrak {D} \; \; \varphi (a) \leqslant \varphi (b) \) or \( \forall a <b \in \mathfrak {D} \; \; \varphi (a) \geqslant \varphi (b) \).
- 3.
At VI International student’s competition on the mathematics of 2012 in Yaroslavl organizers have suggested to study the convergence of this series. It is possible to prove this convergence using some trigonometrical transformations, however little changes of the formula make it impossible to use the solution of a problem in this way (offered by organizers of competitions). Reflections over this problem have led the author to a writing of present article.
- 4.
It denotes, that members of each L-series composing a Z-series decrease more slowly than a geometrical progression with a denominator 0.5. Considering, that Theorem 1 is applied, basically, to conditionally (and very slowly) converging series, such assumption is pertinent.
References
Leibniz, G.W.: De vera proportione circuli ad quadrarum circumpscriptum in numeris rationalibus. Acta Eruditprum, 41–46 (1682)
Lejeune Dirichlet, P.G., Dedekind, R.: Vorlesungen über Zahlentheorie. Brunswick, Lake Forest (1863)
Knopp, K.: Theory and Application of Infinite Series. Dover Publications, New York (1990)
Vorobiev, N.N.: Theory of Series. Nauka, Moscow (1979). (in Russian)
Zverkina, G.A.: On a generalization of the Leibniz theorem. Vestnik TvGU. Seriya: Prikladnaya matematika [Herald Tver State Univ. Ser. Appl. Math.], 2, 123–138 (2014). (in Russian)
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Zverkina, G. (2017). Some Extensions of Alternating Series Test and Its Applications. In: Rykov, V., Singpurwalla, N., Zubkov, A. (eds) Analytical and Computational Methods in Probability Theory. ACMPT 2017. Lecture Notes in Computer Science(), vol 10684. Springer, Cham. https://doi.org/10.1007/978-3-319-71504-9_36
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