Power series convergence

As long as l/i I < 1, this ratio is less than unity and the series Gi and G2 converge. However, for /r = 1 and ju = — I, this ratio equals unity and neither of the infinite power series converges. For the solutions to equation (G.31) to be well-behaved, we must terminate the series Gi and G2 to polynomials by setting... 

This power series converges for po- < 1. Clearly, when pa > 1, the average length per particle p Ms smaller than a, and this is impossible in this system. 

Figure 3.5 The same as Fig. 3.4, except the absorbing potential strength parameter is increased from A = 1.1 eV to 1.5 eV. In this case the optimal T = 4000 au — 97 fs. The power series converges more rapidly with a stronger absorbing potential.

Long-range forces are most conveniently expressed as a power series in Mr, the reciprocal of the intemiolecular distance. This series is called the multipole expansion. It is so connnon to use the multipole expansion that the electrostatic, mduction and dispersion energies are referred to as non-expanded if the expansion is not used. In early work it was noted that the multipole expansion did not converge in a conventional way and doubt was cast upon its use in the description of long-range electrostatic, induction and dispersion interactions. However, it is now established [8, 9, 10, H, 12 and 13] that the series is asymptotic in Poincare s sense. The interaction energy can be written as... 

A power series may be differentiated term by term and represents the function df[x)fdx within the same region of convergence... 

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

In the preceding section, we have established the importance of the power series q x) r(x), 5(x), t x) in combinatorics. Here we examine their analytical properties radius of convergence, singularities on the circle of convergence, analytic continuation. We derive these characteristics from the functional equations whose solutions these series present. I start with a summary of the equations and some notations. 

The power series associated with the functional equations (4.1) - (4.10) have all nonzero radii of convergence. For equations... 

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

Since the coefficients of the power series r(x) are all positive, the point X = p must be singular thus, equation (4.25) holds for x = p. The coefficients of the power series on the left-hand side of (4.25) are all positive, except for the constant, which is equal to 0. On the circle of convergence x = p, the absolute value of the series assumes, therefore, its maximum at x = p. We conclude that x p is the only solution of (4.25) on the circle of convergence the point... 

The generality of a simple power series ansatz and an open-ended formulation of the dispersion formulas facilitate an alternative approach to the calculation of dispersion curves for hyperpolarizabilities complementary to the point-wise calculation of the frequency-dependent property. In particular, if dispersion curves are needed over a wide range of frequencies and for several optical proccesses, the calculation of the dispersion coefficients can provide a cost-efficient alternative to repeated calculations for different optical proccesses and different frequencies. The open-ended formulation allows to investigate the convergence of the dispersion expansion and to reduce the truncation error to what is considered tolerable. 

In the limit as 2 —> oo, this ratio becomes p/k, which approaches zero for finite p. Thus, the series converges for all finite values of p. To test the behavior of the power series as p oo, we consider the Taylor series expansion of... 

Power series have already been introduced to represent a function. For example, Eq. (1-35) expresses the function y = sin x as a sum of an infinite number of terms. Dearly, for x < 1, terms in the series become successively smaller and the series is said to be convergent, as discussed below. The numerical evaluation of the function is carried out by simply adding terms until the value is obtained with the desired precision. All computer operations used to evaluate the various irrational functions are based on this principle. 

A power series may be integrated term by term to represent the integral of the function within an interval of the region of convergence. Iffix) =a0 + ape + apt + , then... 

A well behaved function is one that grows slowly enough at infinity so that the integrand is null. Since the PDF typically falls off exponentially at infinity, any function that can be expressed as a convergent power series will be well behaved. 

The inverse operator may exist even in the case when the corresponding power series in terms of Beffective Hamiltonian, acting within the model space, in the same way as in the multi-root theory, i.e. 

We first consider the case of a two-component solution (biopolymer + solvent) over a moderately low range of biopolymer concentrations, i.e., C < 20 % wt/wt. The quantities pm x in the equations for the chemical potentials of solvent and biopolymer may be expressed as a power series in the biopolymer concentration, with some restriction on the required number of terms, depending on the steepness of the series convergence and the desired accuracy of the calculations (Prigogine and Defay, 1954). This approach is based on simplified equations for the chemical potentials of both components as a virial series in biopolymer concentration, as developed by Ogston (1962) at the level of approximation of just pairwise molecular interactions ... 

In fact, it is too much to require the convergence in the mathematical sense of the formal series of p.m. In all practical problems only their first several terms are calculated, and they may ultimately be divergent, or even possessed of only finite terms, without being unavailable. Thus we are lead to regard them as asymptotic rather than power series, and it is natural to expect that the range of applicability of p.m. be much extended by this new interpretation. 

These convergence radii are in general smaller than that of E% (the latter was shown to be not less than r0> see 2. 2), for the analytic functions (2. 14) may have non-real branch points. But this does not occur if s =l, i.e., if no splitting of the eigenvalue takes place. In this case the power series expressing fa is shown to bo convergent at least for k < n,4 and this is also the case with Ew Ex. 

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