Series Convergent

In case the curve y = fix) is symmetrical with respect to the origin, the a s are all zero, and the series is a sine series. In case the curve is symmetrical with respect to the y axis, the fc s are all zero, and a cosine series results. (In this case, the series will be valid not only for values of x between — c and c, but also for x = — c and x = c.) A Fourier series can always be integrated term by term but the result of differentiating term by term may not be a convergent series. 

There are also two types of convergent series. Consider the new series... 

In this case the series (3-69) is defined as a conditionally convergent series, if the replacement series of absolute values also converges, the series is defined to converge absolutely. 

Two series may be added or subtracted term by term provided each is a convergent series. The joint sum is equal to the sum (or difference) of the individuals. 

The sum of two divergent series can be convergent. Similarly, the sum of a convergent series and a divergent series must be divergent. 

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. 

This criterion allows the computation of the radius of convergence p by means of convergent series. The numerical evaluation requires an estimate of the remainder terms of the series r(x). Relations (10) and (12) provide the inequalities... 

Expanding (7.64) in a convergent series, one can sum over m in the zeroth and first order of expansion (which turns out to be zero) and present the result, accurate to second order in Vxc, as... 

The energy levels of a hydrogen atom are defined by the principal quantum number, n = 1,2, and form a converging series, as shown in Fig. 1.28. 

Convergent series, in resonance theory. 70 Cyanines, absorption of, in relation with pKa, 73... 

If f x) is twice differentiable and satisfies the nonhomogeneous boundary conditions /(O) = /(/) = 0, it arranges itself into a uniformly convergent series... 

This is (at variance with C.3) a rapidly converging series for h Tr/s/a. 

The aldol reaction that establishes two chiral centers in one step has been applied to the synthesis of the ansa chain 66 by Kishi s group as discussed above. Seven chiral centers out of the eight present in the corresponding 66a can be constructed in a different way through a convergent series of four... 

Friedman (1962) has used the cluster theory of Mayer (1950) to derive equations which give the thermodynamic properties of electrolyte solutions as the sum of convergent series. The first term in these series is identical to and thus confirms the Debye-Huckel limiting law. The second term is an I2.nl term whose coefficient is, like the coefficient in the Debye-Huckel limiting law equation, a function of the charge type of the salt and the properties of the solvent. From this theory, as well as from others referred to above, a higher order limiting law can be written as... 

The case of protein synthesis, i.e., diffusion of many segments on the same one-dimensional lattice, is clearly recognized as a species of traffic problem. However, the existing traffic literature has been of no help to its solution. Indeed, only rather special types of solutions have presented themselves to date. On the other hand, methods for a full solution of the DNA synthesis problem, i.e., single-particle diffusion, are reasonably well known but not of much use, since they give solutions in terms of only slowly converging series. Nevertheless, the first two moments of the distribution of degrees of polymerization in the ensemble at each time are easily obtained. 

Fortunately, the Madelung constant may be obtained mathematically from a converging series, and there are computer programs that converge rapidly. However, we need not delve into these procedures, but may simply employ the values obtained by other workers (Table 4.1). The value of the Madelung constant is determined... 

Stefan gave an exact solution for the constant-velocity melting of a semi-infinite slab initially at the fusion temperature. This was extended by Pekeris and Slichter (P2) to freezing on a cylinder of arbitrary surface temperature and Kreith and Romie (K6) to constant-velocity melting of cylinders and spheres by a perturbation method, in which the temperature is assumed to be expressible in terms of a convergent series of unknown functions. To make the method clear, consider the freezing of an infinite cylinder of liquid, of radius r0, at constant surface heat flux. For this geometry the heat equation is... 

Since both J and K integrals depend upon the spatial overlap between the ground state He+ wavefunction and the Rydberg n(m wavefunction, they decrease very rapidly with increasing i and can be expressed as rapidly converging series.3... 

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