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Think Positive! (Simple Math Inequality)
Sum of all matrix elements of n X n Hilbert Matrix M(i,j)=1/(i+j-1) (i,j= 1..n) is equal to HM(n) = 2n*H'(2n), where H'(2n) = H(2n) - H(n), H'(n) = Sum[(1/k)*(-1)^(k+1),(k,1,n}] is an alternate signs Harmonic number, H(n) = Sum[1/k,(k,1,n}] is a Harmonic Number.So, sum of all matrix elements M(i,j) = 1/(i+j) (i,j= 1..n) is equal toHM(n+1) - H(2n+1) = (2n+2)*H(2n+2) - 2(n+1)*H(n+1) - H(2n+1).