Compulsion of a linear equation system to the development of analytic formulas for the sumsof some finite series with the help of special computer programming
Pages 181-186
The article presents a convincing system of mathematical reasoning allowing us to pass over the stages of recurrent formulas as well as the induction methods in the pro- cess of developing analytic formulas using computer programs. The article elaborates the ideas on how to make the computer derive analytic formulas. The author offers us a generalization consisting in using the method of summing up to the more wide range of series, as well as finding approximate specific solutions to some differential equations and summarizations, which can occur, for example, in finite element method. The suggested method of summing the degrees with the coefficient is generalized to:a) The total formulas for the powers degrees of real numbers which are not the rational numbers. This will lead to approximate results.b) The representation of sums is connected to the solutions of certain differential equations (Cauchy problem), where we can obtain the partial equations in the form of power series with rational coefficients.
DOI: 10.22227/1997-0935.2014.1.181-186
- Vladimir Lenev. One of the Methods of How to Make the Computer Derive Analytic Formulas. 14th International Conference on Computing in Civil and Building Engineering. Moscow, June 27—28, 2012, pp. 168—170.
- Lenev V.S. Vyvod formul, vyrazhayushchikh tochno summu nekotorykh konechnykh ryadov s pomoshch'yu EVM [The Development of the Formulas Precisely Expressing Some Finite Series Sums with the help of ECM]. Voprosy prikladnoy matematiki i vychislitel'noy mekhaniki: sbornik nauchnykh trudov [The Collection of Scientific Works: Issues of Applied Mathematics and Computational Mechanics]. Moscow, MGSU Publ., 2000, no. 3, pp. 105—108.
- Lenev V.S. Metod polucheniya s pomoshch'yu EVM klassicheskikh formul dlya ischisleniya konechnykh summ nekotorykh chislovykh ryadov s ispol'zovaniem programmy resheniya v ratsional'nykh chislakh sistemy lineynykh uravneniy razmernosti nxn [Computer-Aided Method for Obtaining Classical Formulas for Numerical Series Sums Using Programs in Rational Numbers in Linear Equation System with the Dimension nxn]. Fundamental'nye nauki v sovremennom stroitel'stve: sbornik dokladov 3-ya nauchno-prakticheskaya konferentsiya [The Collection of Papers (3rd Scientific Conference): Fundamental Sciences in Presentday Construction]. Moscow, 2004, pp. 3—9.
- Brown W.S., Hearn A.C. Applications of Symbolic Algebraic Computation. Computer Physic Communications. 1979, vol. 17, no. 1—2, pp. 207—215.
- Kheming R.V. Chislovye metody [Numerical Methods]. Moscow, Nauka Publ., 1970.
- Akimov P.A., Zolotov A.B., Shirinskiy V.I. Metody tochnogo analiticheskogo resheniya mnogotochechnykh kraevykh zadach stroitel'noy mekhaniki [Methods of Accurate Analytical Solution of Multipoint Boundary Value Problems in Structural Mechanics]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2006, no. 3, pp. 29—39.
- Akimov P.A., Mozgaleva M.L. Korrektnye algoritmy mnogourovnevoy approksimatsii s ispol'zovaniem diskretnogo bazisa Khaara chast' 2: dvumernyy sluchay [Correct Algorithms of Multilevel Approximation Using Discrete Basis of Haar Part 2: Two Dimensional Case]. International Journal for Computational Civil and Structural Engineering. 2012, vol. 8, no. 2, pp. 40—46.
- Munro N., Tsapekis P. Some Recent Results Using Symbolic Algebra. IEE International Conference on Control 94.1994.
- Cohen J.S. Computer Algebra and Symbolic Computation: Elementary Algorithms. AKPeters, LTD, 2002, 323 p.
- Alefeld G., Rohn J., Rump S.M., Yamamoto T. (Eds). Symbolic Algebraic Methods and Verification Methods. Springer, 2001, 266 p.
- Grandshteyn N.S., Ryzhik I.M. Tablitsa integralov, summ, ryadov i proizvedeniy [Table of Integrals, Sums, Series and Products]. Moscow, Nauka Publ., 1971.