Series convergence tests

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. 

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... 

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... 

When the FMS method was first introduced, a series of test calculations were performed using analytical PESs. These calculations tested the numerical convergence with respect to the parameters that define the nuclear basis set (number of basis functions and their width) and the spawning algorithm (e.g., Xo and MULTISPAWN). These studies were used to validate the method, and therefore we refrained from making any approximations beyond the use of a... 

A number of tests are available for confirming the convergence, or otherwise, of a given series. The test for absolute convergence is the simplest, and is carried out using the ratio test. 

There are several tests that will usually tell us whether an infinite series converges or not. 

If r < 1, the series converges. If r > 1, the series diverges. If r = 1, the test fails, and the series might either converge or diverge. If the ratio does not approach any limit but does not increase without bound, the test also fails. 

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. 

Before the general solution can be stated, it is important to determine if the series obtained in Equation 3.74 and Equation 3.75 are convergent. One way is to apply a convergence test[14]. For example, the ratio test... 

All Fourier series have to be made finite when performed numerically the choice of the number of waves used in any calculation is a compromise between the computational effort and the errors caused by the truncation they are difficult to estimate and one usually resorts to numerical testing. An example of a convergence test for a, B - calculated from p(a) as described above (Fig. 3.1) - is s iown in Tab. 3.1 the behavior of truncation errors is typical for many similar situations. Whereas the absolute values of both pressure and energy vary considerably with increasing number of waves, the a, B calculated from them evolve only slowly. Apparently a large part of the truncation error is systematic. Detailed convergence tests for the different potentials used can be found e.g. in Ref. 24 so far the most detailed study... 

We can determine the value of x for which a series converges or diverges by applying the ratio test. We find that the series converges for all values of x in the interval... 

Comparison Tests. The first way to test whether a series converges or diverges is to compare it to one of the series, called comparison series, listed below. Consider the following series ... 

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. ... 

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