I asked:

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