Infinite series definitions

To give a few examples of this failure, we recall the experimental fact that a hydrogen atom, for example, emits an infinite series of sharp spectral lines. Now the hydrogen atom possesses only a single electron, which revolves round the nucleus. By the rules of electrodynamics, an electron accelerated like this sends out radiation continuously, and so loses energy in its orbit it wovdd therefore necessarily get nearer and nearer the nucleus, into which it would finally plunge. The electron, which initially revolved with a definite frequency, will radiate light of this frequency in the case when the... 

Herbert B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed., Macmillan Co., New York, 1962. This book is out of print, but if you can find a used copy, you will find that it is a very useful compilation of formulas, including trigonometric identities, derivatives, infinite series, and definite and indefinite integrals. 

When the sum of an infinite series approaches closer and closer to some definite finite value, as the number of terms is increased... 

The first is that when the formal expressions for thermodynamic properties and correlation functions in the canonical and grand canonical ensembles are converted into graphical language, this definition (or a closely similar one if one chooses to use labeled graphs) is a very natural one to adopt. The second point is that, although the definition is clumsy, the theoretical manipulation of infinite series of such functions is elegant and simple and hence very powerful. 

In this section we first give some statistical-mechanical definitions of the fundamental quantities that are needed for the cluster theory of fluids and some of the elementary relationships among them. Then we will give formulas for the most important of these in terms of infinite series of graphs. 

Infinite series, 94-109 comparison, 95 convergence of, 94-99 tests for, 94-99 definition of, 94 divergence of, 94-99 tests for, 94-99 Fourier, 101-106 Maclaurin, 99 power. See Power series Taylor, 99... 

Recall from calculus the Taylor series expansions for sin t and co , t around t = 0. calculate for each nonnegative integer n. Using the definition of exp as an infinite sum, find an expression for cxptM in terms of sin t and cos t. 

In the preceding section, the continued fractions were introduced in an indirect way into the spectral analysis. Alternatively, they are analyzed in this section in a direct manner. By definition, the infinite-order CF to the series (52) is given by ... 

The mentioned uniqueness of the Fade approximant for the given input Maclaurin series (4) represents a critical feature of this method. In other words, the ambiguities encountered in other mathematical modelings are eliminated from the outset already at the level of the definition of the PA. Moreover, this definition contains its "figure of merit" by revealing how well the PA can really describe the function G(z ) to be approximated. More precisely, given the infinite sum G(2 ) via Eq. (4), the key question to raise is about the best agreement between from Eq. (2) and G(z ) from... 

How are these relations to be interpreted Suppose first that we know the wave numbers with infinite accuracy. The major problem involves the choice of N, which will change the definitions of the c. (We deal with integer m we cannot even think of defining the c in terms of derivatives. Our power series are not mathematicians usual power series.) Clearly, the hope is that we are to choose N so large as not to make its choice affect the values of the c. Two points are now vital ... 

These results of Bruns have been supplemented by Poincare s investigations 1 these lead to the following conditions Apart from special cases, it is not possible to represent strictly the motion of the perturbed system by means of convergent /-fold Fourier series in the time and magnitudes Jk constant in time, which could serve for the fixation of the quantum states. For this reason it has hitherto been impossible to carry out the long-sought-for proof of the stability of the planetary system, i.e. to prove that the distances of the planets from one another and from the sun remain always within definite finite limits, even in the course of infinitely long periods of time. 

Note that subscripts on column matrices designate different matrices, X, X2, etc., whereas Xi designates the ith element of X. Although the Taylor series is infinite, close to extrema we expect a quadratic form to be adequate i.e., for X = X, where X, designates a stationary point and by definition is characterized by f(X ) = 0,... 

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