The Geometric Series

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. 

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]. 

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 ... 

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

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 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 ... 

Turning back to the focus of our interest here, the area under the Normal curve, the statement that the terms of the geometric series never vanish to zero can be reinterpreted in this context as saying that the curves of the Normal curve never intercept the x-axis. Despite this statement, an adaptation of the proof just provided shows that the area under the Normal curve is indeed precisely equal to 1, or 100%. The visual equivalent of this is that there is indeed a defined area under... 

The sums in the brackets are those of the geometric series of operators (matrices). If A is small enough they can be summed up ... 

The series (5.24) can be easily resummed on employing the properties of the geometrical series ... 

The last equality was obtained by defining k = N— 1. The sum of the geometric series... 

Here, x is the sampling or dwell time, whereas ty, 4 are the nodal angular frequencies and the associated amplitudes, respectively By inserting (5) into G(z ) from Eq. (4), the infinite sum over n can be carried out using the exact result for the geometric series = V(1 -Zt/z) = z/(z -Zi). The... 

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. 

The form of the geometric series in equation (1.20) generalizes to the form of equation (1.21) where, now ... 

Instead of writing a series in the form used up to now, in which various terms are exhibited, we can use a standard symbol for a sum, a capital Greek sigma, which we introduced in Chapter 5. For example, the geometric series can be written as... 

EXERCISE 6.5 Show that the geometric series converges if < 1. Q... 

Example.—Show that the series l + + + +. ..is convergent by comparison with the geometrical series l + f + +. .. 

Therefore the character of this operation (a trace of the transformation matrix) is the sum of the geometric series, i.e. 

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