The first two limits of the following problem were proposed at VJIMC, 2005, Category I. The exact value of the last limit was proposed by a user at https://artofproblemsolving.com , where you can also see his solution along other lines. Let (x_n)_{n\ge 2} be a sequence of real numbers, such that x_2>0 and for every n\ge 2 holds x_{n+1}=-1+\sqrt[n]{1+nx_n}. Prove consecutively that 1)\ \lim_{n\to\infty}x_n=0,\ \ \ 2)\ \lim_{n\to\infty}nx_n=0,\ \ \ 3)\ \lim_{n\to\infty}n^2x_n=4. \textbf{Proof.} \textbf{1)} Clearly all the elements of the sequence are positive. The inequality -1+\sqrt[n]{1+nx_n}<x_n is equivalent to (1+nx_n)<(1+x_n)^n, which is seen to be true after expanding, since all the summands on the right are positive. This shows that the sequence is strictly decreasing. Hence 0<x_{n+1}=-1+\sqrt[n]{1+nx_n}\le -1+\sqrt[n]{1+nx_2}, and since the right hand side clearly tends to 0 we obtain $\displaystyle\lim_...
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