Trying to derive Bose-Einstein Condensation using the canonical ensemble

Question

My approach:

1. Write down the partition function $Z_1$ of a single particle,
   approximate the summation with integral so $Z_1 = \int e^{-\beta E} g(E) dE$ where $g(E)$ is the density of states.
2. Mark the ground state with a dummy variable $\mu$ so $Z_1'=Z_1-1+\mu$.
3. The partition function $Z$ of $N$ bosons is $\frac{1}{N!}Z_1^{'N}$.
4. The number of particles in the ground state is $\frac{\partial}{\partial \mu}\log Z\big |_{\mu=1}$.

However, my result is quite different from BEC, specifically, the ratio of particles in the ground state is independent of the particle number $N$.

I suspect this derivation must contain some logical mistakes but I am unable to find them. Any help is appreciated.

Answer 1

The mistake is that the partition function of N bosons can not be written as the multiplication of N partition functions of a single boson because they are indistinguishable.