Absolute convergence

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. 

The expectation value exists in the continuous case only if the integral is absolutely integrable and in the discrete case only if the sum converges absolutely. For statistical samples, Xj, with weights, W , the expectation of g is given by ... 

It can be shown that the series (3-69) does converge to the value S = log 2. However, if each term is replaced by its absolute value, the series becomes unbounded and therefore divergent (unbounded divergent) ... 

Comparison Test. A series will converge if the absolute value of each term (with or without a finite number of terms) is less than the corresponding term of a known convergent series. Similarly, a positive series is divergent if it is termwise larger than a known divergent series of positive terms. 

Alternating-Series Leibniz Test. If the terms of a series are alternately positive and negative and never increase in value, the absolute series will converge, provided that the terms tend to zero as a limit. 

The absolute precision of ERS therefore depends on that of da/dfl (Ej, (t>). Unfortunately, some disagreement prevails among measurements of the recoil cross section. One recent determination is shown in Figure 4a for (t> = 30° and 25°. The convergence of these data with the Rutherford cross section near 1 MeV lends support to their validity. The solid lines are least squares fits to the polynomial form used by Tirira et al.. For (t> = 30°, the expression reads ... 

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... 

Since the elimination method reciuires several operations at one node, the total number of which is independent of the grid step, the algorithm just established will be economical if one succeeds in showing that scheme (9)-(14) is absolutely stable. The following sections place a special emphasis on stability and convergence of the scheme concerned. 

In the early 1990s, Moore et al. reported the syntheses of a tremendous variety of meffl-connected PAMs [5 b, 36]. They recognized that the structural rigidity of systems like 8 could be useful in supramolecular chemistry. The convergent, stepwise approach of linear oligomeric phenylacetylene sequences developed in the Moore laboratory permitted absolute control over chain length, order of... 

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