1/3+1/8+1/15+...+1/120=?

1/3+1/8+1/15+...+1/120=?
1/3+1/8+1/15+...+1/120=?

1/3+1/8+1/15+...+1/120=?

原式＝1/(1*3)+1/(2*4)+1/(3*5)+.....+1/(10*12)=1/2[(1-1/3）+(1/2-1/4)+(1/3-1/5)+......+(1/10-1/12)]=1/2(1-1/3+1/2-1/4+1/3-1/5+........+1/10-1/12)=1/2[（1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+1/10）-(1/3+1/4+1/5+1/6+1/7+1/8+1/9+1/10+1/11+1/12)]=1/2(1+1/2-1/11-1/12)=175/264

1/3=(1-1/3)/2
1/8=(1/2-1/4)/2
以此类推，得到上式=（1-1/3+1/2-1/4+1/3-1/5+1/4-1/6+1/5-1/7+1/6-1/8+1/7-1/9+1/8-1/10+1/-1/11+1/10-1/12)/2=(1+1/2-1/11-1/12)/2=175/264