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# A polynomial root can be a sequence.

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.

Note by Haroun Meghaichi
3 years, 3 months ago