[ METHODES ANALYTIQUES D’AMELIORATION DE LA CONVERGENCE DES SERIES NUMERIQUES REELLES ]
Volume 23, Issue 3, June 2018, Pages 324–340
Théodore Mapendo Wendo1
1 Département de Mathématique-physique, Institut Supérieur Pédagogique d’Idjwi (ISP-IDJWI), Idjwi, Sud-Kivu, RD Congo
Original language: French
Copyright © 2018 ISSR Journals. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
To study the convergence of a real digital series; it calculates the sum Sn of the first n terms. Therefore( lim)(n→∞)〖S_n〗 is calculated (if this limit exists and is equal to S, then the series converges and it converges to S; if this limit doesn’t exist or is infinite, then the series diverges). In practice, it is difficult to calculate Sn for some series. In wanting to get around that, mathematicians have developed convergence criteria deciding on the convergence of the series without calculating the sum. Such is the case of D'Alembert, Cauchy, Riemann, RAABE, DUHAMEL, Gauss ... When a series is recognized convergent, we calculate the approximate sum: the series is converging slowly (for its sum precisely, it is necessary take a large number of words), the series is rapidly convergent (for its sum precisely, take a small number of terms). The transition from slow convergence to the rapid convergence is a numerical analysis problem. So, in this article we would like to get this problem of forgetting. The improvement of the convergence of digital series is obtained from certain transformations using various methods.
Author Keywords: KUMMER’S method, Euler’s method, Krylov’s method, Trigonometrics series.
Volume 23, Issue 3, June 2018, Pages 324–340
Théodore Mapendo Wendo1
1 Département de Mathématique-physique, Institut Supérieur Pédagogique d’Idjwi (ISP-IDJWI), Idjwi, Sud-Kivu, RD Congo
Original language: French
Copyright © 2018 ISSR Journals. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
To study the convergence of a real digital series; it calculates the sum Sn of the first n terms. Therefore( lim)(n→∞)〖S_n〗 is calculated (if this limit exists and is equal to S, then the series converges and it converges to S; if this limit doesn’t exist or is infinite, then the series diverges). In practice, it is difficult to calculate Sn for some series. In wanting to get around that, mathematicians have developed convergence criteria deciding on the convergence of the series without calculating the sum. Such is the case of D'Alembert, Cauchy, Riemann, RAABE, DUHAMEL, Gauss ... When a series is recognized convergent, we calculate the approximate sum: the series is converging slowly (for its sum precisely, it is necessary take a large number of words), the series is rapidly convergent (for its sum precisely, take a small number of terms). The transition from slow convergence to the rapid convergence is a numerical analysis problem. So, in this article we would like to get this problem of forgetting. The improvement of the convergence of digital series is obtained from certain transformations using various methods.
Author Keywords: KUMMER’S method, Euler’s method, Krylov’s method, Trigonometrics series.
Abstract: (french)
Pour étudier la convergence d’une série numérique réelle; on en calcule la somme Sn de n premiers termes, on calcule lim(n→∞)〖S_n〗 (si cette limite existe et est égale à S, alors la série converge et elle converge vers S ; si cette limite n’existe pas ou est infinie, alors la série diverge). Dans la pratique, il est difficile de calculer Sn pour certaines séries. En voulant contourner cette difficulté, les mathématiciens ont élaboré des critères de convergence : on décide de la convergence de la série sans en calculer la somme. Tel est le cas de D’ALEMBERT, CAUCHY, RIEMANN, RAABE, DUHAMEL, GAUSS,…Lorsqu’une série est reconnue convergente, on en calcule la somme approchée : la série est lentement convergente (pour obtenir sa somme avec précision, il faut prendre un grand nombre de termes), la série est rapidement convergente (pour obtenir sa somme avec précision, il faut prendre un petit nombre des termes). Le passage de la convergence lente à la convergence rapide est un problème d’analyse numérique. Ainsi, dans cet article nous voudrions faire sortir ce problème de l’oubli. L’amélioration de la convergence des séries numériques est obtenue à partir de certaines transformations utilisant diverses méthodes.
Author Keywords: méthode de KUMMER, méthode d’Euler, méthode de Krylov, séries trigonométriques.
How to Cite this Article
Théodore Mapendo Wendo, “ANALYTICAL METHODS OF AMELIORATION OF CONVERGENCE OF REELS NUMERICALS SERIES,” International Journal of Innovation and Applied Studies, vol. 23, no. 3, pp. 324–340, June 2018.
