User:Bob K31416/sandbox1

In order to find an expression to simplify the denominator for the case of $$ N \gg 1$$, we start by expanding $$ln \, Z_s(N-1) $$ using a relation for derivatives,


 * $$f(a+h) \simeq f(a) + f'(a) h \qquad \qquad for \ small \ h $$

so that,

Cauchy's estimate of remainder
If there exists a positive real constant Mn such that |&fnof;(n+1)(x)| &le; Mn for all x &isin; (a − r, a + r), then


 * $$ f(x) = f(a) + \frac{f'(a)}{1!}(x - a) + \frac{f^{(2)}(a)}{2!}(x - a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x - a)^n + R_n(x),$$

where the remainder function Rn satisfies the inequality (known as Cauchy's estimate):


 * $$ |R_n(x)| \le M_n \frac{r^{n+1}}{(n+1)!}$$

for all x &isin; (a − r, a + r).

If there exists a positive real constant M1 such that |&fnof;  (x)| &le; M1 for all x'' &isin; (a − r, a + r), then


 * $$ f(x) = f(a) + f'(a) \, (x - a) + R_1(x),$$

where the remainder function R1 satisfies the inequality (known as Cauchy's estimate):


 * $$ |R_1(x)| \le M_1 \frac{r^{2}}{2}$$

for all x &isin; (a − r, a + r).