Some Convergent Constant Series

If a series is to represent a function f x) in some interval, it must be convergent in the entire interval and must converge to the value of the function for every value of X in the interval. For a fixed value of x, the series s(x) is no different from a constant series, and all the tests for convergence of Section 6.1 can be applied. We can then consider different fixed values of x and determine the interval of convergence, that interval in which the series is convergent. 

The series 6 - 12/21/2 + 8/31/2 - 6/41/2 + 24/51/2 -. .. eventually becomes convergent and gives the value for the Madelung constant for the sodium chloride lattice (the standard description of lattices which have the same form as that adopted by sodium fluoride). The values of Madelung constants for some common crystal lattices are given in Table 7.5. 

For three-dimensional crystals the lattice summation converges only slowly, and in any brute-force computational scheme one must make sure that, as one sums outward from the "zeroth unit cell" at the center of the crystal, the ions included at any stage should have as close to zero net charge as possible. For some crystals the Madelung constants a have been evaluated (Table 8.4), using component potentials obtained by summing certain infinite series. 

The second term in parentheses is an alternating series that converges to some value a, known as the Madelung constant. Evaluation of this constant. 

Clearly, we can force a convergence of the A -, if we adjust the AX, appropriately. However, this is not what is intended. Instead, we must check if the quotient in Eq. (3.40) as a function of temperature and the further variable is such that the series in Eq. (3.38) converges. Alternatively, we start with a relation that is used to derive the adiabatic gas expansion. We set up the energy at some constant mol number (dn = 0) either as a function of temperature and volume U T, V) or as a function of entropy and volume U S, V). Both functions must be equal, if the corresponding arguments are inserted into the function declaration ... 

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