Test convergent series

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. 

Rule (iii) is particularly important in the tests for series convergence that will be described in Section 2.11. 

Tacoma Narrows bridge % tangent 16 Taylor s series 32-34 tests of series convergence 35-36 thermodynamics applications 56-57, 81 first law 38-39 Jacobian notation 160-161 systems of constant composition 38 three-dimensional harmonic oscillator 125-128... 

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. 

Observe that 7 - 7 < 0 and consequently AT) > 0 in our notation. If the end temperature is zero (7), = 0) and it cannot be reached in a finite number of steps, then the right-hand side in Eq. (3.38) must be an infinite series that converges to a finite value. This is another formulation of the principle of infinite steps. There are various standard tests for convergence. One of the simplest tests is the ratio test. The series in Eq. (3.38) will converge, if... 

To determine the values of x, which lead to convergent series, we can apply the ratio test (Boas 1983), which states that if the absolute value of the ratio of the (n + 1) term to nth term approaches a limit e as n ao, then the series itself converges when e < 1 and diverges when s > 1. The test fails if e = 1. In the case of the Power Series, Eq. 3.15, we see... 

This is a very slowly convergent series, and provides a severe test of the applicability of LMOs to systems with highly delocalized n systems. 

One of the most sensitive tests of the dependence of chemical reactivity on the size of the reacting molecules is the comparison of the rates of reaction for compounds which are members of a homologous series with different chain lengths. Studies by Flory and others on the rates of esterification and saponification of esters were the first investigations conducted to clarify the dependence of reactivity on molecular size. The rate constants for these reactions are observed to converge quite rapidly to a constant value which is independent of molecular size, after an initial dependence on molecular size for small molecules. The effect is reminiscent of the discussion on the uniqueness of end groups in connection with Example 1.1. In the esterification of carboxylic acids, for example, the rate constants are different for acetic, propionic, and butyric acids, but constant for carboxyUc acids with 4-18 carbon atoms. This observation on nonpolymeric compounds has been generalized to apply to polymerization reactions as well. The latter are subject to several complications which are not involved in the study of simple model compounds, but when these complications are properly considered, the independence of reactivity on molecular size has been repeatedly verified. 

The convergence or divergence of an infinite series is unaffected by the removal of a finite number of finite terms. This is a trivial theorem but useful to remember, especially when using the comparison test to be described in the subsection Tests for Convergence and Divergence. ... 

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. 

Partial Sums of InBriite Series, and How They Grow Calculus textbooks devote much space to tests for convergence and divergence of series that are of little practical value, since a convergent... 

The ANN was able to assimilate the cause-effect relationship of the density of the ester, its structure and temperature. The training and testing results are shown in Fig. 10-14 for individual ester series. The network with the proposed training routine converged in less than 100 iterations for all the esters. 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

The interval of convergence for each of the series solutions u and ui may be determined by applying the ratio test. For convergence, the condition... 

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... 

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