# Summing a general power series

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.

• There is no general way and int he special case you have IO don't think there is a closed form for the sum. Apr 18, 2019 at 5:07