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.

#Calculus #Sequences #Limits

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$