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Your first equation should be a sum over all microstates. If $i$ represents the microstate then the equation is fine. If $i$ represents the energies, then that form is assuming there is no degeneracy in the energy for different microstates. In general, if there are $n_i$ number of microstates with energy $E_i$, then your partition function will be $$Z=\...


Assuming that the power series $$\sum_{s=0}^{\infty}g(N,s)t^s $$ converges for $|t|< T$ and some $T>0$, we can derive it under the sign of the series, therefore: \begin{align} \frac{1}{n!}\left( \frac{d}{dt}\right)^n \sum_{s=0}^{\infty}g(N,s)t^s = \sum_{s=0}^{\infty}g(N,s)\frac{1}{n!}\left( \frac{d}{dt}\right)^n t^s = \sum_{s=n}^{\infty}g(N,s)\frac{...

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