In this paper, we study the relationship between the convergence of the sinusoidal series and the infinity integrals (any real number α ∈[0,1], parameter p > 0). First of all, we study the convergence of the seri...In this paper, we study the relationship between the convergence of the sinusoidal series and the infinity integrals (any real number α ∈[0,1], parameter p > 0). First of all, we study the convergence of the series (any real number α ∈[0,1], parameter p > 0), mainly using the estimation property of the order to obtain that the series diverges when 0 p ≤1-α, the series converges conditionally when 1-α p ≤1, and the series converges absolutely when p >1. In the next part, we study the convergence state of the infinite integral (any real number α ∈[0,1], parameter p > 0), and get that when 0 p ≤1-α, the infinite integral diverges;when 1-α p ≤1, the infinite integral conditionally converges;when p >1, the infinite integral absolutely converges. Comparison of the conclusions of the above theorem, it is not difficult to derive the theorem: the level of and the infinity integral with the convergence of the state (any real number α ∈[0,1], the parameter p >0), thus promoting the textbook of the two with the convergence of the state requires the function of the general term or the product of the function must be monotonically decreasing conditions.展开更多
In this paper, we first discuss the methods of comparing two special absolutely convergentsine series, sinnx and sinnx. We state the theorem in.one dimensional case as follows; Theorem. Let be convergent series with ... In this paper, we first discuss the methods of comparing two special absolutely convergentsine series, sinnx and sinnx. We state the theorem in.one dimensional case as follows; Theorem. Let be convergent series with nonnegative terms. SupposeThen for all x∈[0,π]If, in addition, then展开更多
The Boltzmann equilibrium distribution is an important rigorous tool for determining entropy, since this function cannot be measured, but only calculated in accordance with Boltzmann＇s law. On the basis of the commen...The Boltzmann equilibrium distribution is an important rigorous tool for determining entropy, since this function cannot be measured, but only calculated in accordance with Boltzmann＇s law. On the basis of the commensuration coefficient of discrete and continuous similarly-named distributions developed by the authors, the article analyses the statistical sum in the Boltzmann distribution to the commensuration with the improper integral of the similarly-named function in the full range of the term of series of the statistical sum at the different combination of the temperature and the step of variation （quantum） of the particle energy. The convergence of series based on the Cauchy, Maclaurin criteria and the equal commensuration of series and improper integral of the similarly-named function in each unit interval of variation of series and similarly-named function were estab- lished. The obtained formulas for the commensuration coefficient and statistical sum were analyzed, and a general expres- sion for the total and residual statistical sums, which can be calculated with any given accuracy, is found. Given a direct calculation formula for the Boltzmann distribution, taking into account the values of the improper integral and commensuration coefficient. To determine the entropy from the new expression for the Boltzmann distribution in the form of a series, the conver- gence of the similarly-named improper integral is established. However, the commensuration coefficient of integral and series in each unit interval turns out to be dependent on the number of the term of series and therefore cannot be used to determine the sum of series through the improper integral. In this case, the entropy can be calculated with a given accuracy with a corresponding quantity of the term of series n at a fixed value of the statistical sum. The given accuracy of the statistical sum turns out to be mathematically identical to the fraction of particles with an energy exceeding a given level of the energy barrier equal to the act展开更多
文摘In this paper, we study the relationship between the convergence of the sinusoidal series and the infinity integrals (any real number α ∈[0,1], parameter p > 0). First of all, we study the convergence of the series (any real number α ∈[0,1], parameter p > 0), mainly using the estimation property of the order to obtain that the series diverges when 0 p ≤1-α, the series converges conditionally when 1-α p ≤1, and the series converges absolutely when p >1. In the next part, we study the convergence state of the infinite integral (any real number α ∈[0,1], parameter p > 0), and get that when 0 p ≤1-α, the infinite integral diverges;when 1-α p ≤1, the infinite integral conditionally converges;when p >1, the infinite integral absolutely converges. Comparison of the conclusions of the above theorem, it is not difficult to derive the theorem: the level of and the infinity integral with the convergence of the state (any real number α ∈[0,1], the parameter p >0), thus promoting the textbook of the two with the convergence of the state requires the function of the general term or the product of the function must be monotonically decreasing conditions.
文摘 In this paper, we first discuss the methods of comparing two special absolutely convergentsine series, sinnx and sinnx. We state the theorem in.one dimensional case as follows; Theorem. Let be convergent series with nonnegative terms. SupposeThen for all x∈[0,π]If, in addition, then
文摘The Boltzmann equilibrium distribution is an important rigorous tool for determining entropy, since this function cannot be measured, but only calculated in accordance with Boltzmann＇s law. On the basis of the commensuration coefficient of discrete and continuous similarly-named distributions developed by the authors, the article analyses the statistical sum in the Boltzmann distribution to the commensuration with the improper integral of the similarly-named function in the full range of the term of series of the statistical sum at the different combination of the temperature and the step of variation （quantum） of the particle energy. The convergence of series based on the Cauchy, Maclaurin criteria and the equal commensuration of series and improper integral of the similarly-named function in each unit interval of variation of series and similarly-named function were estab- lished. The obtained formulas for the commensuration coefficient and statistical sum were analyzed, and a general expres- sion for the total and residual statistical sums, which can be calculated with any given accuracy, is found. Given a direct calculation formula for the Boltzmann distribution, taking into account the values of the improper integral and commensuration coefficient. To determine the entropy from the new expression for the Boltzmann distribution in the form of a series, the conver- gence of the similarly-named improper integral is established. However, the commensuration coefficient of integral and series in each unit interval turns out to be dependent on the number of the term of series and therefore cannot be used to determine the sum of series through the improper integral. In this case, the entropy can be calculated with a given accuracy with a corresponding quantity of the term of series n at a fixed value of the statistical sum. The given accuracy of the statistical sum turns out to be mathematically identical to the fraction of particles with an energy exceeding a given level of the energy barrier equal to the act
