Series Geometrical

Consider the sum of a finite number of terms in the geometric series (a special case of a power series). 

In this form, the geometric series is assumed finite. 

The integrals in Eqs. (17) and (18) are called convolution integrals. In Fourier space they are products of the Fourier transforms of c r). Thus, Eq. (18) is a geometric series in Fourier space, which can be summed. Performing this summation and returning to direct space, we have the OZ equation... 

For the parameters used to obtain the results in Fig. 3, X 0.6 so the mean free path is comparable to the cell length. If X -C 1, the correspondence between the analytical expression for D in Eq. (43) and the simulation results breaks down. Figure 4a plots the deviation of the simulated values of D from Do as a function of X. For small X values there is a strong discrepancy, which may be attributed to correlations that are not accounted for in Do, which assumes that collisions are uncorrelated in the time x. For very small mean free paths, there is a high probability that two or more particles will occupy the same collision volume at different time steps, an effect that is not accounted for in the geometric series approximation that leads to Do. The origins of such corrections have been studied [19-22]. 

Although the hypergeometric functions are useful in spectroscopy, as they describe the rotation of a symmetric top molecule (Section 9.2.4), their importance is primarily due to their generality. If, for example, a = 1 and fi say, Eq. (154) becomes a +i — a for all values of n. The result is the ordinary geometric series... 

This a geometric series. When the last term goes to zero,... 

This is a geometric series with ratio a/ 3. Introducing its sum results in... 

It should be noted that, if we adopt a certain convention, a hypergeomctric series can stop and start ngnin after a number of zero terms. For example, consider the hyper-geometric series a V—11 b — n —mi ) where both m and... 

This rather awkward expression can be simplified by recognizing the presence of a geometric series ... 

This expression corresponds to a summation over a geometric series, so that... 

Summation of the geometric series over all species for the total concentration and rearrangement gives... 

Most contemporary schemes for designing atomic and molecular Gaussian basis sets e.g. [4]- [7]) exploit, in one form or another, the idea of even-tempered basis sets [8], [9] (see also [4]), which have exponents, (p, defined by the geometric series ... 

Detailed description of the domains of convergence of hyper geometric series in terms of amoeba of the discriminant of the polynomial has been given recently in Passare and Tsikh (2004). The discriminant A(a) is an irreducible polynomial with integer coefficients in terms of the coefficients , of polynomial (54) that vanishes if this polynomial has multiple roots. For instance, for cubic polynomial the discriminant is... 

Mark Z. Lazman and Gregory S. Yablonsky, Overall Reaction Rate Equation of Single-Route Complex Catalytic Reaction in Terms of Hyper geometric Series... 

In the Layer Doubling method the diffraction properties of pairs of layers are determined exactly from those of the individual layers this is done by summing up the multiple scattering between the layers as in a geometrical series, but using matrix inversion rather than the series expansion. By repeating this combination of layers, the... 

The sum in Eq. (10.26) is well known as a convergent geometric series, so that we may write... 

Finally, with the help of Eq. 5.88, by summing the geometric series over n, we get the line shape according to... 

Let us now consider more closely the replacement of the summations by integrals. The summations (3.20) are simple enough to be evaluated rigorously, In fact they can be reduced to a geometric series. Some algebra yields... 

In order to estimate the transcendental number e, we will expand the exponential function ex in a power series using a simple iterative procedure starting from its definition Eq. (25) together with Eq. (12). As a prelude, we first find the power series expansion of the geometric series y — 1/(1 + x), iterating the equivalent expression ... 

The sum of the diagonal elements of the MR r( ) of the symmetry operator R(o z) forms a geometric series which we have summed before in Section 7.2 with j replaced by /, an integer. As before,... 

The infinite summation in this expression is the geometrical series. The infinite summation (S ) of a geometrical series... 

The solution is related to the observation that the sum of an infinite series can converge to a finite solution. An example that effectively demonstrates the solution here is the geometric series 1/2 + 1/4 + 1/8 +. .. ad infinitum. That is, the series starts with 1/2, and every subsequent term is one half of the previous term. Given this, the terms of the series never vanish to zero. However, the sum of them is precisely 1. The proof of this is as follows, where the series is represented as 5 ... 

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