Let us consider the following function: $$f(s_1,s_2,...s_n,n)=\frac{1}{1+s_1}+\frac{1}{(1+s_2)^2}+...+\frac{1}{(1+s_n)^n},$$ where $s_i\in(0,1), i=1,...,n.$ Let us introduce the following function: $$g(x,n)=\frac{1}{1+x}+\frac{1}{(1+x)^2}+...+\frac{1}{(1+x)^n}.$$ We require that $f(s_1,s_2,...s_n,n)=g(x,n)$ and from this condition it is needed to find $x$. In other words it is needed to construct $x(s_1,...,s_n)$. Since $n$ attains large numbers (e.g. 25) according Abel-Ruffini theorem this problem isn't possible to solve analytically.
Please guide me how to find some approximation. Does iteration method a good way to solve the problem? (I mean saying iteration following: assign to $s_i$'s values, find $f(s_1,...s_n)$, therefore estimate $x$).