Let v_n be the vector in R^(n-1) defined as (1/n, 2/n,...,(n-1)/n).
Let w_n be the vector in R^(n-1) with the same coordinates in reverse order.
Find the limit as n -> oo of the angle between v_n and w_n.
(Try this without using numerical approximation or summation-of-powers formulae.)
Is there a simple geometric reason for the answer? I don't know.
--Dan
Ok, I cheated (numerical approximation) and found limiting angle pi/4 (45 degrees). I don't know a simple geometric reason for this, but perhaps someone can use the following to find one. Permute the coordinates of v_n (or w_n). All such points lie in an (n-2)-plane, and their convex hull forms a "permutahedron". Permutahedra tile Euclidean space. I consider them generalized hexagons (the 3-dimensional member is the truncated octahedron). v_n and w_n are diametrically opposite points of the permutahedron, so v_n - w_n is twice its circumradius. A normal to the permutahedron hyperplane through the origin intersects the permutahedron center c_n = (m/n, m/n, ..., m/n) where m = n/2. Does some relationship between permutahedra circumradii and the distance between the origin and the hyperplace containing the permutahedron give any insight? - Scott