Flip some sum

Level pending

Let

\(\begin{equation} f(x) = 1+\frac{2}{x}+\frac{3}{x^2}+\frac{4}{x^3}+\dots\end{equation}\)

If \(f(-101)=S\), for what value of \(x\) does \(f(x)=\frac{1}{S}\)?

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