The Principle of Infinite Steps

Observe that 7 - 7 0 and consequently AT) 0 in our notation. If the end temperature is zero (7), = 0) and it cannot be reached in a finite number of steps, then the right-hand side in Eq. (3.38) must be an infinite series that converges to a finite value. This is another formulation of the principle of infinite steps. There are various standard tests for convergence. One of the simplest tests is the ratio test. The series in Eq. (3.38) will converge, if 

More straightforward in the present case is the integral test. If the terms of the series are replaced by a continuous function f x) that would emerge, when k is replaced by X in the formula for AT t with 

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  

We annotate that the consideration can be formulated more generally, using a general extensive variable X instead of the volume V. In this case, Eq. (3.41) would look like 

Since we are familiar with the volume energy we do not use the general equation, but exemplify our consideration with the volume energy as given in Eq. (3.41), even when in practice magnetic effects play a major role to achieve deep temperatures. By equating the inner equation of the set of equations in Eq. (3.41), we get 

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