I was solving a problem and I came across the following expression,
$$\sum_n^N {N \choose n}\exp[-\beta n\omega]$$
I was looking for the convergence of this series but I couldn't find any resources which relates this sum of exponentials with a combinatory factor, so I was wondering if you have seen this series before or know any table where I can find general series for converging exponentials.
This sum is of the form $$ \sum_n \binom{N}{n} a^n $$ where $a = e^{-\beta\omega}$. So by the binomial theorem, this is equal to $$ \sum_n \binom{N}{n} a^n = (1+a)^n $$ and the answer is $$ (1+e^{-\beta\omega} )^n $$