While solving a problem involving equilibrium positions of charges on a line, I came up with a recurrence relation which is nonlinear, and moreover implicitly defined. Here it is: [itex]x_{0}=0[/itex] and [itex]\sum^{n-1}_{i=0} \frac{1}{(x_{n}-x_{i})^{2}} = 1[/itex]. I should also mention that [itex]0 \leq x_{n}< x_{n+1}[/itex] for all [itex]n[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

I can't even find an explicit expression for [itex]x_{n}[/itex] as a function of the previous terms, let alone as a function of [itex]n[/itex]. I've dealt with linear recurrences before, but how would I go about solving a nonlinear recurrence like this? Is it even possible to find a closed-form expression using elementary functions? If an exact solution is impossible, is there some way to get a numerical approximation?

Any help would be greatly appreciated.

Thank You in Advance.

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# How do you solve a nonlinear recurrence relation?

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