Suppose $x_n$ is an infinite sequence of positive numbers less than or equal to $1$ (or perhaps a fixed positive real $r$), whose sum diverges to infinity. Let $P$ be a point in the real plane. We say $P$ is accessible according to $x_n$ iff there is a finite sequence $P_0, P_1, ... , P_n$ such that $P_0$ is the origin, $P_n=P$, and the distance between $P_k$ and $P_{k+1}$ is $x_k$. Then, is every point in the plane accessible according to $x_n$, since we are assuming the sum diverges to infinity? It is certainly a necessary condition for every point to be accessible that the sum be infinite. I am asking whether it is sufficient also.
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- 2$\begingroup$ What have you tried? Where did you get stuck? $\endgroup$David G. Stork– David G. Stork2020-10-21 01:04:07 +00:00Commented Oct 21, 2020 at 1:04
- $\begingroup$ As you are very used to this site, why don't you answer the question ? I have another question : could you say in which more precise context than given by tag "geometry" you are asked to answer this question ? Is it "Computational geometry" ? $\endgroup$Jean Marie– Jean Marie2020-10-22 09:15:01 +00:00Commented Oct 22, 2020 at 9:15
- $\begingroup$ If your divergent series is harmonic series, here is a connected issue: math.stackexchange.com/q/2377994 $\endgroup$Jean Marie– Jean Marie2020-10-22 09:28:37 +00:00Commented Oct 22, 2020 at 9:28
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