Let \(X\) have a geometric distribution with

\[P(X=i)=(1-p) p^{i} ; \quad i=0,1, \ldots ; 0

By mixing these geometric distributions with regard to a suitable structure distribution density \(f(p)\) show that

\[\sum_{i=0}^{\infty} \frac{1}{(i+1)(i+2)}=1\]