简便计算：1*2分之1+2*3分之1+3*4分之1······+38*39分之1+39*40分之1

1/[n*(n+1)]=1/n-1/(n+1)
1*2分之1+2*3分之1+3*4分之1······+38*39分之1+39*40分之1
=1/1-1/2+1/2-1/3+1/3-1/4.+1/38-1/39+1/39-1/40
=1-1/40
=39/40
1/[n*(n+2)]=1/2*[1/n-1/(n+2)]
1*3分之1+3*5分之1+5*7分之1+······+19*21分之1+21*23分之1
=1/2*(1/1-1/3)+1/2*(1/3-1/5).+1/2*(1/19-1/21)+1/2*(1/21-1/23)
=1/2(1/1-1/3+1/3-1/5+.+1/19-1/21+1/21-1/23)
=1/2(1-1/23)
=1/2*22/23
=11/23