Convergence of Series

After 1000 terms, the sum equals 7.485. It might appear that, with sufficient patience, the series will eventually converge to a finite quantity. Not so. 

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Because of this property and of the convergence of series (71), we shall treat all diagrams of the same class in the same way. 

The latter variable eliminates a finite range [42] of convergence of series both of X at 2 i e, because of a pole due to intemuclear coulombic repulsion as / 0, and of y at V2 Re, for a similar phenomenon as oo. Expressions for Fy in terms of coefficients cj in the latter series are available in a large consistent collection in Fortran coding [43] up to Cio such expressions, readily calculated, through symbolic computation with efficient procedures [44], first in terms of coefficients and thence converted to bj or Cj as required, are further converted to Fortran or C code for numerical applications. 

Parameters a,. and bt can be calculated as in [5] and in [12]. They are small and greater than zero. They go to zero as i °° thus, it is possible to ensure the convergence of series. Tinear approximation only requires finding and (or b0 and bt), which are, in case of maximum power output, as follows ... 

We list here some of the tests that can be used to determine the convergence of series without having to evaluate the series ... 

Not much is known concerning the convergence of series occurring in quantum chemistry. Comm iiy, only a few perturbational corrections are computed. 

Proof During the iteration process, the value of Sj changes for different iteration steps. We aim to derive the condition for the convergence of series (22) for an arbitrary value of sj. This will guarantee the convergence of the iterative method, no matter what the value of sj. The complex matrix power series (22) converges if, and only if, for all the eigenvalues crP of the matrix Ry, the inequal-... 

Brooks F C 1952 Convergence of intermolecular force series Phys. Rev. 86 92... 

A potential advantage of methods based on a series expansion of the free energy is that the convergence of the series is determined by the A dependence of the potential energy function meaning that the efficiency of the approach could be enhanced by a judicious choice of coupling scheme. 

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. 

Partial Sums of InBriite Series, and How They Grow Calculus textbooks devote much space to tests for convergence and divergence of series that are of little practical value, since a convergent... 

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... 

Things have moved on since the early papers given above. The development of Mpller-Plesset perturbation theory (Chapter 11) marked a turning point in treatments of electron correlation, and made such calculations feasible for molecules of moderate size. The Mpller-Plesset method is usually implemented up to MP4 but the convergence of the MPn series is sometimes unsatisfactory. The effect... 

The sums in Eqs. (1) and (2) run, respectively, over the reciprocal space lattice vectors, g, and the real space lattice vectors, r and Vc= a is the unit cell volume. The value of the parameter 11 affects the convergence of both the series (1) and (2). Roughly speaking, increasing ii makes slower the convergence of Eq. (1) and faster that of Eq. (2), and vice versa. Our purpose, here, is to find out, for an arbitrary lattice and a given accuracy, the optimal choice, iiopt > tbal minimises the CPU time needed for the evaluation of the KKR structure constants. This choice turns out to depend on the Bravais lattice and the lattice parameters expressed in dimensionless units, on the... 

The values are supposed to be finite. There are no assumptions on the convergence of the power series (I.I) formal expansions... 

In the Introduction, the radii of convergence of the four power series... 

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