Question

1/(1*2)+2/(1*2*3)+3/(1*2*3*4)+4/(1*2*3*4*5)+5/(1*2*3*4*5*6)+6/(1*2*3*4*5*6*7

简便运算

Answer 1

解：通项为k/(k+1)!=(k+1)/(k+1)!-1/(k+1)!=1/k!-1/(k+1)!
令k=1,2,……,7得
1/2!=1-1/2!
2/3!=1/2!-1/3!
……
6/7!=1/6!-1/7!
上面等式相加得S=1-1/7!,

Answer 2

通项为k/(k+1)!=(k+1)/(k+1)!-1/(k+1)!=1/k!-1/(k+1)!
令k=1,2,……,6得
1/2!=1-1/2!
2/3!=1/2!-1/3!
……
6/7!=1/6!-1/7!
上面等式相加得S=1-1/7!,

= 5 040 分之 5039

Answer 3

原式=1+1/2！+1/3！+1/4!+1/5！+1/6！-1/2！-1/3！-1/4！-1/5！-1/6！-1/7！
=1-1/7！
=5039/5040