Impossible sum.

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.

×