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.

The mathematical notion of nonlinear partial derivatives equations are registered in full in Mathematical Analysis. This concept has several applications in other disciplines such as physics, economics, demography, chemistry, differential geometry and infinitesimal, etc. Presented with diversified forms, this notion remains essential in the progress of all research in pure and applied mathematics. It borrows concepts of topology and functional analysis for a better understanding. The question is at what level mathematics are for research and our contribution in this article. For this, we present the theory and develop some methods of solving equations to nonlinear partial differential equations.

In mathematical analysis, methods were developed for solving differential equations as there is no general method for solving these equations. However, in Algebra, the simultaneous equations are those for which the roots all check equations, they form the system of equations. We note in the study of differential equations that the system of differential equations is not very developed yet many books to refer analysis but not explicitly. And these systems can be put in matrix form. This allows the resolution of these systems by using the matrices being given various operations on the matrix and determinant calculation. The ideal of this article is to make the public interested in mathematics, clear text with a more understandable language on solving linear ordinary differential equations systems with constant coefficients. Our research is limited to the case of linear differential equations with constant coefficients. We do not claim to theorize on solving ordinary linear differential equation systems with constant coefficients less and extend applications.