Thread: a question of sequence limit

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

How to solve this ? thanks

I see, it is .5 .

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

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

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

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

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

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$