Series divergence

This means that the discrete solution nearly conserves the Hamiltonian H and, thus, conserves H up to 0 t ). If H is analytic, then the truncation index N in (2) is arbitrary. In general, however, the above formal series diverges as jV —> 00. The term exponentially close may be specified by the following theorem. 

As it stands, the series diverges for any z. It is usually used together with the recursion z = (z + l) /(z + l), which is first employed to push the argument arbitrarily far into the asymptotic region, to provide any requested accuracy of the Stirling formula. 

The above method is formal in the sense that the convergence of the various series is not discussed. Indeed, the series diverge for many applications. However, the lower orders of the transformed system can give interesting information, and the process can be stopped at a certain order M. This means that these terms of the series are useful to construct both the transformed Hamiltonian and the generating function, since they are unaffected by the ultimately divergent character of the whole process. 

Comparison of the accuracies of the Chebyshev and Bernoulli approximations in Table XI shows the Chebyshev (L = 0) to be better than the Bernoulli by a factor of 5 to 10 at n = 1 the improvement increases to about 300 at n = 4. This was generally observed for all other isotopic substitutions tested the rate of convergence of the Chebyshev expansion is better than the Bernoulli expansion at any order, at any temperature. The Chebyshev expansion exists at any temperature, while the Bernoulli series diverges for most of the molecules at room temperature. Table XI also shows that the Bigeleisen-Mayer approximation. 

Many people are surprised when they first learn that this series diverges, because the terms keep on getting smaller as you go further into the series. This is a necessary condition for a series to converge, but it is not sufficient. We will show that the harmonic series is divergent when we introduce tests for convergence. 

The nth-Term Test. If the terms of a series approach some limit other than zero or do not approach any limit as you go further into the series, the series diverges. 

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. 

The exponent n is an important quantity in the empirical description of the interatomic interactions. This quantity uniquely determines the dependence of energy on distance, E r) Series of the type of (15.18) have been calculated and tabulated (see [7]). For n < 3 these series diverge. As n ooA approaches the number of nearest neighbors, which is 12 for the face-centered crystal lattice. 

Fig. 5.3. The complex plane of the A parameter. The physically interesting points are at A = 0,1. In perturbation theory we finally put A = 1. Because of this the convergence radius pi of the perturbational series has to be Pk 1. However, r/any com/)/eJC A tvitfi A < 1 corresponds to a pole of the energy, the perturbational series will diverge in the physieal situation (A = 1). The figure shows the position of a pole by concentric circles, (a) The pole is too close pj < 1) and the perturbational series diverges (b) the perturbational series converges, because p/ > 1.

The range on this expansion is also restricted because the series diverges when... 

Observe that the series diverges for certain values of the variables, producing nonsquare-integrable wavefunctions. Correct this by requiring that the series terminate. This requires that the truncated series be either symmetric or antisymmetric in the variable and also that p of Eq. (4-38) and (4-39) be equal to /(/ - -1) with / an integer. 

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