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Define $a_1=8$ and $a_n=\frac{4^{n+1}-2^{n+2}\sqrt{4^n-a_{n-1}}}{2}$ for $n\geq 2$.

By means of harmonic analysis methods it can be shown that $a_n$ converges to $\pi^2$ (this being the first eigenvalue of for the string of unit length and Dirichlet bc).

Does anyone know how to prove this from a different approach?

More generally, I would like to hear if there is some sort of theory (or theories) dealing with sequences as the one above; to be a bit more precise: what can be said of a sequence defined by $a_n=f_m(a_{n-1})$ where $f_n$ are functions depending on $n$ in a (fairly simple and regular) way.

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I doubt that there is a general theory, but for this particular example there is a different way. Let $b_n = 4^{-(n+1)} a_n$, then it is easy to check that $b_1 = 1/2$ and $b_{n+1} = \frac{1-\sqrt{1-b_n}}{2}$, i.e., that it really boils down to the asymptotic behavior of the iterates of $1/2$ under the function $f(b) = \frac{1-\sqrt{1-b}}{2}$.

Now it is another fortunate coincidence that $f$ happens to be conjugate to the inverse of the Chebyshev polynomial $T(z) = 2 z^2 - 1$ via the change of coordinates $z = 1-2b$. This leads to the identity $b_n = \frac{1-\cos (2^{-n} \pi)}{2}$ and the Taylor expansion $\cos z = 1 - \frac{z^2}{2} + O(z^4)$ gives the desired result.

Obviously both tricks (transforming it into a problem of asymptotic behavior of the iterates of one function, as well as having an explicit expression for high iterates) are very specific to this one example.

On the other hand, if you are just trying to decide whether the sequence is bounded, bounded away from 0, or other questions that involve somewhat coarser asymptotics, there might be some general classes of examples that can be dealt with.

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