Given a series of the form
$$\sum_{r=0}^N f(r)\alpha^r\tag{1}$$
where $f$ is an analytic function of $r$, $N$ is an integer, and $\alpha>0$, how could one solve the above for it's sum?
In particular, I'm interested in the sum for $f(r)=\frac{1}{1+k_1e^{-k_2r}}$, where $k_1$, $k_2 >0$. However, I was wondering if there is a general way to look at such sums by putting some constraints on $f$ such as the one I've put---analyticity.
