Consider Einstein solid model ($N$ oscillators of same frequency $\omega$, where $n=\sum k_i $ with $k_i$ being the occupation number of single oscillators)
In microcanonical ensemble entropy is $$S=k \ln (\frac{(N+n-1)!}{n! (N-1)!}) \sim_{\mathrm{thermodynamic \,\, lim. \,\, (N\to \infty)\,\,\,\,\,}} k \ln (\frac{(N+n)^{N+n}}{n^n (N)^N})\tag{1}$$
On the other hand in canonical ensemble, since the partition function if $Z=(\frac{1}{2\sinh(\beta\omega\hbar/2)})^N$ one finds that $$S=\frac{kN}{2} [\beta\hbar\omega \coth(\beta\omega\hbar/2)-2 \ln(2\sinh(\beta\omega\hbar/2))]\tag{2}$$
Where $\beta=1/kT$.
I would like to show that two expressions are the same in the thermodynamic limit which was not taken for $(2)$.
Firstly I also got this relation between $\beta $ and $n$ $$\beta(n)=\frac{1}{\hbar \omega} \ln (\frac{N}{n}+1) \tag{3}$$
I tried to put $\beta(n)$ in $(2)$ and make an approximation for $N\to \infty $ in $(2)$ but I could not get to $(1)$ in any way.
What is the way to correctly approximate $(2)$ (with $\beta (n)$) to get to $(1)$?