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Given a sequence $(x_n)_{n \in \mathbb{N}^0}$ defined by a quadratic map $$x_{n+1} = ax_n^2 + bx_n + c$$ with $x_0 = 0$ and $a,b,c \in \mathbb{R}$,
is there a fast way to decide whether the sequence converges or not?

So far I've been able to come up with some obvious tests that only can prove divergence, but not convergence, such as if the quadratic map doesn't have a real fixed point $(b^2 < 4ac)$, or if $b\geq1$ and $a,c > 0$, then the sequence diverges.

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  • $\begingroup$ I think the fastest way is drawing the graph of $y=x$ and $y=ax^{2}+bx+c$ and draw the sequence by hand.. $\endgroup$ – Seewoo Lee Mar 1 '18 at 12:50

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