# Impossible sum.

**Number Theory**Level 5

Function \(f:\) \(N\)x\(N\)\(→N\) which satisfy

a. \( f(n,n)=n\) for all natural numbers \(n\).

b. \(f(n,m)=f(m,n)\) for all natural numbers \(n,m\)

c. \(\frac{f(m,n+m)}{f(m,n)}=\frac{m+n}{n}\) for all natural numbers \(n,m\)

Let \(S=\frac{1}{f(1,2)f(3,4)} + \frac{1}{f(2,3)f(4,5)} + \frac{1}{f(3,4)f(5,6)}+....\) upto \(∞\)

Find sum of all possible values of \(S\) correct upto \(3\) decimal places.