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Math Problem in Mathematica

The problem I would like to formulate in Mathematica is: Let f[i,j] =
Abs[Sqrt[1-(i/n)^2]-j/n]. i and j run from  1 to n, and n is a fixed
integer >=1.  I want to find the  sum S of the minimum of f over j, for
each i for given n.

Example: n=7.  Min (i=1, j from 1 to 7)= .01
               Min (i=2, """          )= .04
               Min (i=3, ...           = .05

and the sum S = 0.20.   

FindMinimum seemed like the right idea, but I don't know how to make it
work for a function of discrete values.

The problem appears as Problem 10365 in AMM. 

Daniel Tisdale

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