Testwiki:Reference desk/Archives/Mathematics/2018 November 16

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November 16

Limit of real root of a polynomial

Let $f_n(x)=\sum _{k=0}^n{x}^{2^k-1}$. It's easy to see that there is always 1 real root, call it $r_n$. Does $\underset{n\to \mathrm{\infty }}{\lim }{r}_n$ have a closed form? Is there anything interesting to say about it? Numerically it is approximately -0.658626754300164. 98.190.129.147 (talk) 17:43, 16 November 2018 (UTC)

Apparently the number is mentioned in this article. It's got a pay wall though. --RDBury (talk) 22:03, 16 November 2018 (UTC)

That article credits the sum to [1]. Unfortunately neither says much more about the real zero than you've figured out already (one has a proof of convergence), they're both mostly about complex zeros of $f_n(x)$. 78.0.230.255 (talk) 00:50, 17 November 2018 (UTC)