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

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

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 $\lim _{n\to \infty }r_{n}$ 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)