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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.

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

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