Easy to show that : \(x^n+x^2=1\) has a unique solution in \([0,1]\) for any \(n\in \mathbb{N}\). Let \(x_n\) be that solution.

First you may try to prove that \(x_n\to 1 \), Can you show that \(\frac{n(1-x_n)}{\ln n}\to 1 \) ?

What does the latter limit means ? it means that : \(x_n= 1- \frac{\ln n}{n} + o\left(\frac{\ln n}{n} \right) \), and this can tell us how fast this sequence converges.

You may check this problem.

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