In 1827, mathematician Peter Lejeune-Dirichlet discovered the surprising result that some

convergent series, when rearranged, can yield a different result [3].

When dealing with the absolutely

convergent series [mathematical expression not reproducible], we take into account that

Gautschi's contributions opened the door for extensive work on orthogonal polynomials and their applications in diverse areas of applied and numerical analysis, including numerical integration, interpolation processes, integral equations, moment-preserving spline approxination, and the summation of slowly

convergent series.

Although these methods prove to be effective in solving most of nonlinear differential equations and in obtaining a

convergent series solution, they have few disadvantages such as the large number of terms in the solution as the number of iterations increases.

Their topics include the measurability and semi-continuity of multifunctions, the optimality of function spaces in Sobolev embeddings, a note on the off-diagonal Muckenhoupt-Wheedon conjecture, the Radon-Nikod<'y>m theorem for vector measures and integral representation of operators on Banach function spaces, and the Orlicz-Pettis theorem for multiplier

convergent series. ([umlaut] Ringgold, Inc., Portland, OR)

It provides the solution with high accuracy and minimal calculation in a rapidly

convergent series which lead to the solution in a closed form by using the initial condition only.

By choosing the convergence control parameter value other than optimal (but from the effective region) we get a

convergent series as well, only the rate of convergence of the series will be less.

Homotopy analysis method (HAM) is implemented for obtaining the

convergent series solutions of the transformed equations.

Moreover, by means of the so-called h-curve, a valid region of h can be studied to gain a

convergent series solution.

It is clear that [x.sup.[alpha]] [subset] [X.sup.[beta]] and [X.sup.[alpha]] [subset] [X.sup.[gamma]], but [X.sup.[beta]] [subset] [X.sup.[gamma]] does not hold, since the sequence of partial sums of a double

convergent series need not to be bounded.

Tsalamengas and Fikioris [9] have proposed a technique based on the asymptotic approximation in the space domain followed by rapidly

convergent series [12] to accelerate the summation of series.

the fluid velocity components as sums of

convergent series using the Adomian decomposition technique and compute the admissible values of the shear-stress on the plate surface Consider the stream function [psi]