# Thread: a question of sequence limit

1. ## a question of sequence limit

$\forall\,n\in \mathbb{N}$, $x_n$ 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

2. I see, it is .5 .

3. 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

$\chi$ $\sigma$