I asked: << 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.

...

Scott writes: << Ok, I cheated (numerical approximation) and found limiting angle pi/4 (45 degrees).

...

That's not the answer I get. The cosine of the angle sought is the limit as n --> oo of the dot product f(n) = (v_n / ||v_n||) · (w_n / ||w_n||). Multiplying the numerator and denominator of f(n) each by 1/n, one sees that as n --> oo they approach integral_0^1 x(1-x) dx and integral_0^1 x^2 dx respectively, so the cosine is 1/2, and the angle is pi / 3. --Dan