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Bertrand's theorem (Wikipedia)

Regarding central force motion, Bertrand's theorem states that only inverse-square and linear force law produce stable closed, if bounded, orbits.

Wikipedia's proof of the theorem (Link above) first establishes orbital equation in the form$$\frac{d^2\eta}{d\theta^2}+\beta^2\eta=\hat J(\eta),$$where $\eta=u-u_0=\frac{1}{r}-\frac{1}{r_0}$ and $\hat J$ is a power series of order $\geq2$, and exploits the Fourier expansion of $\eta$ with respect to $\beta \theta$.

Goldstein (Classical Mechanics, 3rd ed, p.90~1) demonstrates only part of Bertrand's theorem. The above equation is solved for small perturbation (small $\eta$ i.e. $\hat J$ is negligible) and the solution $\eta=A\cos\beta \theta$ is given. Then it is argued that $\beta$ must be a rational constant for the orbit to be closed.

But I feel this argument is inaccurate, since $\eta=A\cos\beta\theta$ is only an approximate solution in the first place. What if the actual orbit (which is a solution of the equation above, taking $\hat J$ into account) is not closed even if $\beta$ is rational? Did I make a mistake in this line of thinking?

*Additional question

In Wikipedia's proof, why can one expand $\eta$ with $cos(n\beta\theta)$?

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