Conditionally convergent series

In this case the series (3-69) is defined as a conditionally convergent series, if the replacement series of absolute values also converges, the series is defined to converge absolutely. 

As has been extensively discussed in the literature, these expressions are conditionally convergent series and appropriate procedures have been developed to guarantee their physically correct evaluations [11-15]. Different, though... 

This potential ( )(r) is infinite if the central cell is not neutral, i.e., the sum of qi is not zero, and otherwise is an example of a conditionally convergent infinite series, as discussed above, so a careful treatment is necessary. The potential depends on the order of summation, that is, the order in which partial sums over n are computed. For example, for positive integers K, define ( )s (r) as... 

If f x) is twice differentiable and satisfies the nonhomogeneous boundary conditions /(O) = /(/) = 0, it arranges itself into a uniformly convergent series... 

If a series is conditionally convergent, its sums can be made to have any arbitrary value by a suitable rearrangement of the series it can in fact be made divergent or oscillatory (Riemanns theorem). 

This series of resonance conditions converges to 2.618..., implying that the most important bottleneck to intramolecular energy transfer is determined by a golden mean cantorus, that is,... 

However, it is possible that the two conditions may be equivalent.) For the sake of clarity of discussions in the following sections we assume this stronger condition i.e., existence of the convergent series for F p, and the cell-pair correlation function which can be obtained from F p - Note, however, that this assumption can always be replaced by a (possibly) weaker assumption [for example, the condition (b) for /4 p ] in the ensuing discussion. Similar Taylor expansion for the normalization constant C in (4) can be carried out however, we shall not discuss it here explicitly. Only the final result of the expansion will be presented at the end of the derivation. 

The solution in Equation 8.26 is inconvenient for several reasons. Each term in the series contains two coefficients w and A ) which require numerical calculation. In the case of a linear wall reaction, these quantities depend on the wall kinetic parameter, and this relationship is recently obtained in a simple and explicit manner by Lopes et al. [40]. In addition, whenever this slowly convergent series is used to describe the inlet region, a large number of terms may be required so that a satisfactory result is obtained. The efficient evaluation of the terms in Graetz series has been the object of many studies. Housiadas et al. [51] presented a comparative analysis between several methods to estimate these terms, remarking the numerical issues associated with the rigorous calculation of these quantities. However, this was done for uniform wall concentration (Dirichlet boundary condition), excluding the important case of finite reaction rates. 

The electrostatic potential energy in a charged system can be written as the summation of all pair-wise coulombic interactions between charges. In a periodic array, it also includes the interaction with the infinite number of replica charges generated by the periodic repetition of the simulated system. The series of coulombic terms converges very slowly, and the solution depends on the order of the summation i.e., the series is conditionally convergent. 

Since n can, in principle, assume very large values and since the factor in front of the summation sign in eqn. (1) has a positive value, the above condition can only be satisfied if a is smaller than / . In this case the summation represents a convergent series, yielding an additional condition for dry operation ... 

Lowdin (10) has already stressed the formal difficulties which would arise in the numerical solutions of eq. 35. Furthermore, starting with the work by O Shea and Santry in 1974 (12a) and Ukrainski in 1975 (12b), it became gradually apparent that model chains embody also size related difficulties mathematically expressed as conditionally (Coulombic interactions) and sometimes slowly (exchange contributions) convergent series (lattice sums). The basic implications were also fully appreciated since terms of the series involve multicenter integrals and straightforward summations are prohibitive (12c). 

There is no difficulty in evaluating the repulsive matrix R(q) since the series (4.31) converges rapidly. An example will be given below for NaCl. However, some care is required in evaluating the expression (4.32) because of the long-range nature of the Coulomb potential, the series is only conditionally convergent. 

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