Infinitely many capacitors

Calculus Level 5

\[\large \sum_{n=0}^\infty \left(\int_{r_n}^{r_{n+1}} f(x) \, dx \right)^{-1} = \dfrac1C \]

Let \(r_n\) denote the \(n^\text{th} \) smallest non-negative root of \(f(x) = \sin\sqrt x\), where \({r_0}=0\). Find \(C\).

If you believe that the sum above diverges, write your answer as zero.


Image Credit: Capacitors - 12739s by Eric Schrader, Wikipedia.
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