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Math Help - a question of sequence limit

  1. #1
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    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
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  2. #2
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    I see, it is .5 .
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  3. #3
    MHF Contributor chisigma's Avatar
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    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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