Deriving Fermi-Dirac Distribution misunderstanding

[d2699]

Homework Statement

The actual question was deriving Bose-Einstein, but I got confused on the F-D example. I'm basically following the method given here.

Homework Equations

[All taken directly from the above link]

Taylor series:

The Attempt at a Solution

So after that third equation is where I can't figure it out. To relate Z_s(N-1) to Z_s(N), we Taylor expand Z_s(N-ΔN) to get:

where

Looking specifically at that middle step, I figured the terms after the second term were truncated. I don't understand what is being used as the function, what are x and a, etc. Can someone point me in the right direction?

 

[Charles Link]

$x=N-\Delta N$, and $a=N$. They also take $Z_s=e^{ln(Z_s)}$, and let $\Delta N=1$. And also $e^{a+b}=e^a e^b$