1/m(m+1)+1/(m+1)(m+2)+1/(m+2)(m+3)
=[(m+2)(m+3)+m(m+3)+m(m+1)]/m(m+1)(m+2)(m+3)
=(3m*m+9m+6)/m(m+1)(m+2)(m+3)
=3(m+1)(m+2)/m(m+1)(m+2)(m+3)
=3/m(m+3)