1/3+1/3*5+1/5*7.1/99*101

1/3+1/3*5+1/5*7........1/99*101
=1/(1*3)+1/(3*5)+...+1/(99*101)
=(1/2)*(1/1-1/3)+(1/2)*(1/3-1/5)+...+(1/2)*(1/99-1/101)
=(1/2)*(1/1-1/3+1/3-1/5+...+1/99-1/101)
=(1/2)*(1-1/101)
=(1/2)*(100/101)
=50/101.

1/3=1/2*(1-1/3);
1/(3*5)=1/2*(1/3-1/5);
……
上面式子=1/2*（1-1/3+1/3-1/5……-1/99+1/99-1/101）=1/2*(1-1/101)=50/101

1/3*5=1/2[1/3 -1/5] 1/5*7=1/2[1/5-1/7],......
1/99*101=1/2[1/99-1/101]
1/3+1/3*5+1/5*7........1/99*101=1/3+1/2[1/3 -1/5+1/5-1/7+...+1/99-1/101]=1/3+1/2[1/3-1/101]=49/303

由于1/(2n-1)(2n+1)=[1/(2n-1)-1/(2n+1)]/2
所以1/3+1/3*5+1/5*7........1/99*101
=[(1-1/3)+(1/3-1/5)+...+(1/99-1/101)]/2
=(1-1/101)/2
=50/101

这是裂项法
1/3=（1-1/3）/2
1/3*5=（1/3-1/5）/2
所以都拆了 中间约去只剩首尾
即为（1-1/101）/2=50/101

1/3+1/3*5+1/5*7........1/99*101
=1/3+1/2*(1/3-1/5)+1/2*(1/5-1/7)+......1/2*(1/99-1/101)
=1/3+1/2*1/3-1/2*1/101
=1/3+1/6-1/202=50/101