(i) Verify that \(\frac{2 r+1}{r(r+1)(r+2)}=\frac{1}{2}\left\{\frac{(2 r+1)(2 r+3)}{(r+1)(r+2)}-\frac{(2 r-1)(2 r+1)}{r(r+1)}\right\}\).

(ii) Hence show that \(\sum_{r=1}^{n} \frac{2 r+1}{r(r+1)(r+2)}=\frac{1}{2}\left\{\frac{(2 n+1)(2 n+3)}{(n+1)(n+2)}-\frac{3}{2}\right\}\).

(iii) Deduce the value of \(\sum_{r=1}^{\infty} \frac{2 r+1}{r(r+1)(r+2)}\).