a question of sequence limit

**water2011** · Mar 17th 2010, 04:16 AM

$\forall\,n\in \mathbb{N}$ , $x_n$ is the root of equation $x+x^2+\cdots+x^n=1$ on interval $[0,1]$ , Please solve $\lim\limits_{n\to \infty}x_n$ .

How to solve this ? thanks

**water2011** · Mar 17th 2010, 05:00 AM

I see, it is .5 .

**chisigma** · Mar 17th 2010, 05:35 AM

If we denote with $x_{n}$ the sequence of real roots, is ...

$x_{n}\cdot (1 + x_{n} + \dots + x_{n}^{n-1}) = \frac{x_{n}\cdot (1-x_{n}^{n})}{1-x_{n}} = 1$ , $\forall n$ (1)

But it is also $0 < x_{n} < 1$ so that is...

$\lim_{n \rightarrow \infty} x_{n}^{n} = 0$ (2)

... and, as consequence, if we indicate with $x_{\infty}$ the limit of the $x_{n}$ , is...

$\frac{x_{\infty}}{1-x_{\infty}}= 1 \rightarrow x_{\infty} = \frac{1}{2}$ (3)

Kind regards

$\chi$ $\sigma$

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