Series finite

For any sequence of terms u, uj, 3,. .., we can form a iinite series by summing the terms in the sequence up to and including the nth term  

For example, the sum of the first n terms in the series obtained from the sequence defined by equation (1.8) is given by  

Evaluating this sum for n= 1,2, 3,4, 5 yields the sequence of partial sums  

If we now look closely at this new sequence of partial sums, we may be able to deduce that the sum of the first u terms is 5 = 2 —1. In general, for a geometric series obtained by summing the members of 

For the geometric series obtained by summing the first n terms of the geometric progression in equation (1.8), use equation (1.20) and appropriate values of a and x given in equation (1.10) to confirm that the sum of the first n terms is 2 -1. 

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When one is dealing with a finite series of reactions, it is possible to use stoichiometric principles to determine the concentration of the final species. For example, if only four species (.A, B, C, D) are involved in the sequence of equation 5.3.2, then... 

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

The potential outside a charge distribution can be expressed in terms of a finite series of the outer moments of the distribution. The expression is obtained through a power series expansion of r1, where r is the distance from the field point to the origin of the distribution, and subsequent integration (Hirshfelder et al. 1954, Buckingham 1978). At a point rf, with components ra, for unit value of 47te0, one obtains... 

Enskog obtained a solution by expanding A, (Wj) and 6, (W,) in a finite series of Sonine polynomials [60,114,178]. This solution is usually found to converge in only one or two terms. More discussion of the solution itself can be found in Chapman and Cowling [60] and in Hirschfelder, Curtiss, and Bird [178]. We are concerned here with using the Enskog result to obtain transport coefficients. 

Configuration interaction has come to mean any expansion of the wavefunction in a finite series of N-electron functions (28)... 

When there is more than one electron, the best Rnps are of the SCF (self-consistent-field) type obtainable by the method of Hartree and Fock. These are not given by analytical expressions, and are usually presented in the form of numerical tables. Moreover, these SCF R t s as tabulated are not always all orthogonal however, it is always jxjssible to find an equivalent set of equally good SCF R,u which are orthogonal.2 For practical computations, as Slater has shown,3 the SCF Rni may be approximated passably well by a finite series similar to that of Eq. (4) but with a different exponential factor in each term ... 

Fig. 52. Finite series of nesting spheres representing atoms, repeat units, branch cells, dendrons, dendrimers, dendrimer clusters, and dendrimer macro-lattices (ordered infinite networks) illustrating an abiotic hierarchy analogous to that found in biological system...

The effectiveness factor in terms of /3 and y for an uth-order reaction may be calculated from a finite series for a region of low ji (Tavera, 2005)... 

Where H is the similarity-transformed Hamiltonian, eq (14), with respect to two independent cluster operators T and Z or, more precisely, with respect to the excitation operator T and the deexcitation operator Z The advantage of eq (36) over the expectation value of the Hamiltonian with the CC wave function, which can also improve the results for multiple bond breaking (28, 127), is the fact that EcC(z,j is a finite series in T and Z. Unfortunately, the power series expansions of (Z,7), eq (36), in terms of T and Z contain higher powers of... 

The orthogonal functions s (( 0x,(ay) are given in terms of finite series of products of Hermite polynomials Hn [51] in the components of the angular velocity as... 

Tgj is represented exactly and the exact electronic energy, which also includes dispersion effects correctly, is obtained. However, this comes with infinite computational costs. Hence, methods needed to be devised, which allow us to approximate the infinite expansion in Eq. (12.9) by a finite series to be as short as possible. A straightforward approach is the employment of truncated configuration interaction (CI) expansions. Note that (electronic) configuration refers to the set of molecular orbitals used to construct the corresponding Slater determinant. It is a helpful notation for the construction of the truncated series in a systematic manner and yields a classification scheme of Slater determinants with respect to their degree of excitation . Excitation does not mean physical excitation of the molecule but merely substitution of orbitals occupied in the Hartree-Eock determinant o by virtual, unoccupied orbitals. Within the LCAO representation of molecular orbitals the virtual orbitals are obtained automatically with the solution of the Roothaan equations for the occupied orbitals that enter the Hartree-Eock determinant. 

Although they appear evident, two points that are essential for the further considerations are explicitly mentioned here. First, the finite series resistance that is necessary to destabilize a steady state implies that the double-layer potential (in this chapter always denoted as (tipL) differs considerably from the externally fixed voltage U. We denote as poten-tiostatic conditions operating conditions under which U is fixed but (jipL. as well as the current, may evolve in time. Systems in which the external resistance can be neglected (and hence U = ( )dl) are called strictly potentiostatic. [Obviously, in a strictly potentiostatic system, the doublelayer potential does not represent a variable that can vary with time according to Eq. (1) but reduces to a parameter, and any instability in a strictly potentiostatic system is of a chemical nature.] Second, recall that the equivalent circuit describes a galvanostatic system for Rg o° and [7 00, and thus galvanostatic systems are naturally included in the analysis. 

The summation of a finite series will always yield a finite result, but the summation of an infinite series needs careful examination to confirm that the addition of successive terms leads to a finite result, i.e. the series converges. It is important not to confuse the notion of convergence as applied to a series with that applied to a sequence. For example, the harmonic sequence given by equation (1.14) converges to the limit zero. However, somewhat surprisingly, the harmonic series ... 

The distinction between a finite series having a finite sum and an infinite series where a finite sum exists only if the series converges. 

A series is the snm of a seqnence. Thus, given a sequence, one can form a series by snmming the sequence. A finite seqnence generates a finite series, and an infinite sequence generates an infinite series. 

A sequence is a set of numerical quantities with a rule for generating one member of the set from the previous one. If the members of a sequence are added together, the result is a series. A finite series has a finite number of terms, and an infinite series has a infinite number of terms. If a series has terms that are constants, it is a constant series. Such a series can be written... 

Equation (6.9) is valid for any value of r, since a finite series always converges. 

In this chapter we introduced mathematical series and mathematical transforms. A finite series is a sum of a finite number of terms, and an infinite series is a sum of infinitely many terms. A constant series has terms that are constants, and a functional series has terms that are functions. The two important questions to ask about a constant series are whether the series converges and, if so, what value it converges to. We presented several tests that can be used to determine whether a series converges. Unfortunately, there appears to be no general method for finding the value to which a convergent series converges. 

Another modelling strategy is based on the use of spherical harmonics, where the excess scattering density p(r) - pg is expanded as a finite series of multipoles in order to approximate the shape of the particle [57]. While the experimental and calculated 1(Q) curves can be compared in this way, it has not yet been shown that the particle shape can be readily visuahzed in the way that is possible by computer displays of Debye spheres. 

If V is to constitute a finite series, the coefficient of some power of X in (4) must become zero, so that... 

This series has an infinite number of terms and is therefore caiied an infinite series. A finite series has a fixed number of terms. 

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