Operations with Infinite Series

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.  

If a series is conditionally convergent, its sums can be made to have any arbitrary value by a suitable rearrangement of the series it can in fact be made divergent or oscillatory (Riemanns theorem). 

A series of positive terms, if convergent, has a sum independent of the order of its terms but if divergent, it remains divergent however its terms are rearranged. 

An oscillatory series can always be made to converge by grouping terms. 

A power series can be inverted, provided the first-degree term is not zero. Given 

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Whether this concept can stand up under a rigorous psychological analysis has never been discussed, at least in the literature of theoretical physics. It may even be inconsistent with quantum mechanics in that the creation of a finite mass is equivalent to the creation of energy that, by the uncertainty principle, requires a finite time A2 A h. Thus the creation of an electron would require a time of the order 10 20 second. Higher order operations would take more time, and the divergences found in quantum field theory due to infinite series of creation operations would spread over an infinite time, and so be quite unphysical. 

So, for any fixed N, we obtain the list of elementary ( 3,4,5), 3)-polycycles with N interior faces. The enumeration is done in the following way run the computation up to A = 13 and obtain the sporadic elementary ( 3,4,5, 3)-polycycles and the members of the infinite series. Then undertake, by hand, the operation of addition of a face and reduction by isomorphism we obtain only the 21 infinite series for all N > 13. This completes the enumeration of finite elementary ( 3,4,5), 3)-polycycles. 

The main drawback of the chister-m-chister methods is that the embedding operators are derived from a wavefunction that does not reflect the proper periodicity of the crystal a two-dimensionally infinite wavefiinction/density with a proper band structure would be preferable. Indeed, Rosch and co-workers pointed out recently a series of problems with such chister-m-chister embedding approaches. These include the lack of marked improvement of the results over finite clusters of the same size, problems with the orbital space partitioning such that charge conservation is violated, spurious mixing of virtual orbitals into the density matrix [170], the inlierent delocalized nature of metallic orbitals [171], etc. 

Power series have already been introduced to represent a function. For example, Eq. (1-35) expresses the function y = sin x as a sum of an infinite number of terms. Dearly, for x < 1, terms in the series become successively smaller and the series is said to be convergent, as discussed below. The numerical evaluation of the function is carried out by simply adding terms until the value is obtained with the desired precision. All computer operations used to evaluate the various irrational functions are based on this principle. 

Carrying out mathematical operations such as integration or differentiation on a functional series with a finite number of terms is straightforward, since no questions of convergence arise. However, carrying out such operations on an infinite... 

Thus, Ih Ceo is the first of an infinite magic-number series of neutrals and / , C20 the first of a series with closed-shell dications but open-shell neutrals. Inspection of the two series shows a general geometrical relationship in that any open-shell icosahedral fullerene C with n = 60k + 20 = 20(3k + 1) canbe converted into a larger icosahedral fullerene C3 with 3n = 60(3k + 1) by a specific transformation, and all icosahedral closed-shell neutrals are so produced. The conversion operation is the so-called... 

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