I have trouble to find a closed-form expression for finite series. Before it, let's take into consideration the following very straightforward example: $$S_{n}=\frac{1}{1+r}+\frac{1}{(1+r)^2}+...+\frac{1}{(1+r)^n}.$$ It's clear that $$S_n=\frac{1}{r} \cdot \left[1-\frac{1}{(1+r)^n}\right]$$.
Now let's assume that $r$ isn't constant, in other words, let's assume that we have the following sequence: $$r_1, r_2, ...,r_n.$$ Therefore the $S_n$ is given by $$S_n=\frac{1}{1+r_1}+\frac{1}{(1+r_2)^2}+...+\frac{1}{(1+r_n)^n}.$$ Is it possible to express above mention by a close-form expression? If yes, how?
P.S. One can assume relationship between {$r_i$}s, $i=1,...n$.