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Series infinite

We can also form an infinite series from a sequence by extending the range of the dummy index to an infinite number of terms  [Pg.7]

The summation of a finite series will always yield a finite result, but the summation of an infinite series needs careful examination to confirm that the addition of successive terms leads to a finite result, i.e. the series converges. It is important not to confuse the notion of convergence as applied to a series with that applied to a sequence. For example, the harmonic sequence given by equation (1.14) converges to the limit zero. However, somewhat surprisingly, the harmonic series  [Pg.7]

each successive sum of terms in parentheses will always be greater than 2 For example  [Pg.7]

In Chapter 1 of Volume 1 we saw that the irrational number, n, can be calculated from the sum of an infinite series. One example given involved the sum of the inverses of the squares of all positive integers  [Pg.8]

This series converges extremely slowly, requiring well over 600 terms to provide precision to the second decimal place in order to achieve 100 decimal places for ti, we would need more than 10 terms However, the alternative series  [Pg.8]


Now let us eonsider a funetion that is periodie in time with period T. Fourier s theorem states that any periodie funetion ean be expressed in a Fourier series as a linear eombination (infinite series) of Sines and Cosines whose frequeneies are multiples of a... [Pg.548]

Simplifying this result involves the same infinite series that we examined in connection with Eq. (5.27) therefore we can write immediately... [Pg.452]

Those involving series truncation. The quantity In (1 - X2) can be represented by the infinite series - [x2 + (1/2) x + (1/3) x - - ]. Truncating this series after the first term is a valid approximation for dilute solutions and also simplifies the form of the equation. It is an optional step, however, and can be avoided or mitigated by simply retaining more terms in the series. [Pg.546]

The electromagnetic field of a light beam produces an electrical polari2ation vector in the material through which it passes. In ordinary optics, which may be termed linear optics, the polari2ation vector is proportional to the electric field vector E. However, the polari2ation can be expanded in an infinite series ... [Pg.12]

The volumetric properties of fluids are conveniently represented by PVT equations of state. The most popular are virial, cubic, and extended virial equations. Virial equations are infinite series representations of the compressibiHty factor Z, defined as Z = PV/RT having either molar density, p[ = V ), or pressure, P, as the independent variable of expansion ... [Pg.484]

If n is not a positive integer, the sum formula no longer apphes and an infinite series results for (a + by. The coefficients are obtained from the first formulas in this case. [Pg.431]

In the form of Eq. (3-67), it can further be defined that the terms in the series be nonending and therefore an infinite series. [Pg.448]

For this, it is stated the infinite series converges if the limit of approaches a fixed finite value as n approaches infinity. Otherwise, the series is divergent. [Pg.448]

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. ... [Pg.449]

Since the initial condition must be satisfied, we use an infinite series of these functions. [Pg.458]

Virial Equations of State The virial equation in density is an infinite-series representation of the compressiDility factor Z in powers of molar density p (or reciprocal molar volume V" ) about the real-gas state at zero density (zero pressure) ... [Pg.529]

Application of an infinite series to practical calculations is, of course, impossible, and truncations of the virial equations are in fact employed. The degree of truncation is conditioned not only by the temperature and pressure but also by the availability of correlations or data for the virial coefficients. Values can usually be found for B (see Sec. 2), and often for C (see, e.g., De Santis and Grande, ATChP J., 25, pp. 931-938 [1979]), but rarely for higher-order coefficients. Application of the virial equations is therefore usually restricted to two- or three-term truncations. For pressures up to several bars, the two-term expansion in pressure, with B given by Eq. (4-188), is usually preferred ... [Pg.529]

These equations have been solved analytically for solid slabs, cylinders, ana spheres. The solutions are in the form of infinite series, and usually the results are plotted as curves involving four ratios [Gurney and Lurie, Jnd. Eng. Chcm., 15, 1170 (1923)] defined as follows with q = 0-. [Pg.556]

A. Laminar, vertical wetted wall column Ws/, 3.41 — D 5fa (first term of infinite series) [T] Low rates M.T Use with log mean concentration difference. Parabolic velocity distribution in films. [Pg.607]

Although the differential equation is first-order linear, its integration requires evaluation of an infinite series of integrals of increasing difficulty. [Pg.695]

Gram-Charlier Series This is an infinite series whose coefficients involve the Gaussian distribution and its derivatives (Kendall, Advanced Theory of Statistics, vol. 1, Griffin, 1958). The derivatives, in turn, are expressed in terms of the moments. The series truncated at the coefficient involving the fourth moment is... [Pg.2086]

This potential ( )(r) is infinite if the central cell is not neutral, i.e., the sum of qi is not zero, and otherwise is an example of a conditionally convergent infinite series, as discussed above, so a careful treatment is necessary. The potential depends on the order of summation, that is, the order in which partial sums over n are computed. For example, for positive integers K, define ( )s (r) as... [Pg.106]

This infinite series converges rapidly, and evaluation with N varying from - 4 to + 4 is usually sufficient, These equations are used when evaluating by computer, as the series 3 can easily be evaluated. [Pg.299]

Now, the quadrupole moment can next be calculated by differentiating the potential to get the electric field due to the dipole moment. The reader can now see that an infinite series can be thus generated. The total electric field is simply the sum of all the individual multipole contributions, given by... [Pg.166]

Solutions to Fourier s equation are in the form of infinite series but are often more conveniently expressed in graphical form. In the solution the following dimensionless groups are used. [Pg.391]

The propagator may thus be written as an infinite series of expectation values of increasingly complex operators over the reference wave function. [Pg.259]

This equation is the first term of an infinite series which appears in the rigorous solution of the quasi-diffusion. This equation describes the regular process of quasi-diffusion. For the low values of the Fourier number (irregular quasi-diffusion) it is necessary to use Eq. (5.1) or Boyd-Barrer approximation [105, 106] for the first term in Eq. (5.1)... [Pg.39]

The last theorem will not be proved, but it will be used to prove the following important theorem the infinite series... [Pg.56]

Notice that the sign of i is not changed in this transposition of Hn. Thus, when the infinite series is restored, expression (7-7) becomes... [Pg.393]

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. [Pg.450]

Analytical solutions of equation 9.44 in the form of infinite series are available for some simple regular shapes of particles, such as rectangular slabs, long cylinders and spheres, for conditions where there is heat transfer by conduction or convection to or from the surrounding fluid. These solutions tend to be quite complex, even for simple shapes. The heat transfer process may be characterised by the value of the Biot number Bi where ... [Pg.401]

A solution to Equation (8.12) together with its boundary conditions gives a r, z) at every point in the reactor. An analytical solution is possible for the special case of a first-order reaction, but the resulting infinite series is cumbersome to evaluate. In practice, numerical methods are necessary. [Pg.271]

To identify the governing processing and material parameters, a one dimensional case was analyzed. The heat transfer problem renders an exact solution, [10], which can be presented as an infinite series... [Pg.126]

This analytic solution for [l]° involves evaluation on an infinite series for G(t). However, for the application studied, the series converges rapidly and five terms were sufficient for accurate results. [Pg.309]


See other pages where Series infinite is mentioned: [Pg.202]    [Pg.10]    [Pg.10]    [Pg.549]    [Pg.286]    [Pg.419]    [Pg.419]    [Pg.431]    [Pg.448]    [Pg.448]    [Pg.449]    [Pg.449]    [Pg.468]    [Pg.97]    [Pg.397]    [Pg.364]    [Pg.17]    [Pg.29]    [Pg.85]    [Pg.65]    [Pg.404]    [Pg.71]   
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See also in sourсe #XX -- [ Pg.106 ]

See also in sourсe #XX -- [ Pg.159 ]

See also in sourсe #XX -- [ Pg.119 ]

See also in sourсe #XX -- [ Pg.10 ]




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